Matrices and Matrix Operations
Learning Objectives
By the end of this section, you will be able to:- Find the sum and difference of two matrices.
- Find scalar multiples of a matrix.
- Find the product of two matrices.
Figure 1. (credit: "SD Dirk," Flickr)| Wildcats | Mud Cats | |
|---|---|---|
| Goals | 6 | 10 |
| Balls | 30 | 24 |
| Jerseys | 14 | 20 |
Finding the Sum and Difference of Two Matrices
To solve a problem like the one described for the soccer teams, we can use a matrix, which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named [latex]A,B,\text{}[/latex] and [latex]C[/latex] are shown below.[latex]A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{ccc}1& 2& 7\\ 0& -5& 6\\ 7& 8& 2\end{array}\right],C=\left[\begin{array}{c}-1\\ 0\\ 3\end{array}\begin{array}{c}3\\ 2\\ 1\end{array}\right][/latex]
Describing Matrices
A matrix is often referred to by its size or dimensions: [latex]\text{ }m\text{ }\times \text{ }n\text{ }[/latex] indicating [latex]m[/latex] rows and [latex]n[/latex] columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix [latex]A[/latex] identified as [latex]{a}_{ij},\text{}[/latex] we look for the entry in row [latex]i,\text{}[/latex] column [latex]j[/latex]. In matrix [latex]A\text{, \hspace{0.17em}}[/latex] shown below, the entry in row 2, column 3 is [latex]{a}_{23}[/latex].[latex]A=\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right][/latex]
A square matrix is a matrix with dimensions [latex]\text{ }n\text{ }\times \text{ }n,\text{}[/latex] meaning that it has the same number of rows as columns. The [latex]3\times 3[/latex] matrix above is an example of a square matrix.
A row matrix is a matrix consisting of one row with dimensions [latex]1\text{ }\times \text{ }n[/latex].
[latex]\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right][/latex]
A column matrix is a matrix consisting of one column with dimensions [latex]m\text{ }\times \text{ }1[/latex].
[latex]\left[\begin{array}{c}{a}_{11}\\ {a}_{21}\\ {a}_{31}\end{array}\right][/latex]
A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations.
A General Note: Matrices
A matrix is a rectangular array of numbers that is usually named by a capital letter: [latex]A,B,C,\text{}[/latex] and so on. Each entry in a matrix is referred to as [latex]{a}_{ij}[/latex], such that [latex]i[/latex] represents the row and [latex]j[/latex] represents the column. Matrices are often referred to by their dimensions: [latex]m\times n[/latex] indicating [latex]m[/latex] rows and [latex]n[/latex] columns.Example 1: Finding the Dimensions of the Given Matrix and Locating Entries
Given matrix [latex]A:[/latex]- What are the dimensions of matrix [latex]A?[/latex]
- What are the entries at [latex]{a}_{31}[/latex] and [latex]{a}_{22}?[/latex]
[latex]A=\left[\begin{array}{rrrr}\hfill 2& \hfill & \hfill 1& \hfill 0\\ \hfill 2& \hfill & \hfill 4& \hfill 7\\ \hfill 3& \hfill & \hfill 1& \hfill -2\end{array}\right][/latex]
Solution
- The dimensions are [latex]\text{ }3\text{ }\times \text{ }3\text{ }[/latex] because there are three rows and three columns.
- Entry [latex]{a}_{31}[/latex] is the number at row 3, column 1, which is 3. The entry [latex]{a}_{22}[/latex] is the number at row 2, column 2, which is 4. Remember, the row comes first, then the column.
Adding and Subtracting Matrices
We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries. In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add or subtract a [latex]\text{ }3\text{ }\times \text{ }3\text{ }[/latex] matrix and another [latex]\text{ }3\text{ }\times \text{ }3\text{ }[/latex] matrix, but we cannot add or subtract a [latex]\text{ }2\text{ }\times \text{ }3\text{ }[/latex] matrix and a [latex]\text{ }3\text{ }\times \text{ }3\text{ }[/latex] matrix because some entries in one matrix will not have a corresponding entry in the other matrix.A General Note: Adding and Subtracting Matrices
Given matrices [latex]A[/latex] and [latex]B[/latex] of like dimensions, addition and subtraction of [latex]A[/latex] and [latex]B[/latex] will produce matrix [latex]C[/latex] or matrix [latex]D[/latex] of the same dimension.[latex]A+B=C\text{ such that }{a}_{ij}+{b}_{ij}={c}_{ij}[/latex]
[latex]A-B=D\text{ such that }{a}_{ij}-{b}_{ij}={d}_{ij}[/latex]
Matrix addition is commutative.
[latex]A+B=B+A[/latex]
It is also associative.
[latex]\left(A+B\right)+C=A+\left(B+C\right)[/latex]
Example 2: Finding the Sum of Matrices
Find the sum of [latex]A[/latex] and [latex]B,\text{}[/latex] given[latex]A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\text{ and }B=\left[\begin{array}{cc}e& f\\ g& h\end{array}\right][/latex]
Solution
Add corresponding entries.[latex]\begin{array}{l}A+B=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]+\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{ccc}a+e& & b+f\\ c+g& & d+h\end{array}\right]\hfill \end{array}[/latex]
Example 3: Adding Matrix A and Matrix <>B
Find the sum of [latex]A[/latex] and [latex]B[/latex].[latex]A=\left[\begin{array}{cc}4& 1\\ 3& 2\end{array}\right]\text{ and }B=\left[\begin{array}{cc}5& 9\\ 0& 7\end{array}\right][/latex]
Solution
Add corresponding entries. Add the entry in row 1, column 1, [latex]{a}_{11},\text{}[/latex] of matrix [latex]A[/latex] to the entry in row 1, column 1, [latex]{b}_{11}[/latex], of [latex]B[/latex]. Continue the pattern until all entries have been added.[latex]\begin{array}{l}A+B=\left[\begin{array}{cc}4& 1\\ 3& 2\end{array}\right]+\left[\begin{array}{cc}5& 9\\ 0& 7\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{ccc}4+5& & 1+9\\ 3+0& & 2+7\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{cc}9& 10\\ 3& 9\end{array}\right]\hfill \end{array}[/latex]
Example 4: Finding the Difference of Two Matrices
Find the difference of [latex]A[/latex] and [latex]B[/latex].[latex]A=\left[\begin{array}{cc}-2& 3\\ 0& 1\end{array}\right]\text{ and }B=\left[\begin{array}{cc}8& 1\\ 5& 4\end{array}\right][/latex]
Solution
We subtract the corresponding entries of each matrix.[latex]\begin{array}{l}A-B=\left[\begin{array}{rr}\hfill -2& \hfill 3\\ \hfill 0& \hfill 1\end{array}\right]-\left[\begin{array}{rr}\hfill 8& \hfill 1\\ \hfill 5& \hfill 4\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{rrr}\hfill -2 - 8& \hfill & \hfill 3 - 1\\ \hfill 0 - 5& \hfill & \hfill 1 - 4\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{rrr}\hfill -10& \hfill & \hfill 2\\ \hfill -5& \hfill & \hfill -3\end{array}\right]\hfill \end{array}[/latex]
Example 5: Finding the Sum and Difference of Two 3 x 3 Matrices
Given [latex]A[/latex] and [latex]B:[/latex]- Find the sum.
- Find the difference.
[latex]A=\left[\begin{array}{rrr}\hfill 2& \hfill -10& \hfill -2\\ \hfill 14& \hfill 12& \hfill 10\\ \hfill 4& \hfill -2& \hfill 2\end{array}\right]\text{ and }B=\left[\begin{array}{rrr}\hfill 6& \hfill 10& \hfill -2\\ \hfill 0& \hfill -12& \hfill -4\\ \hfill -5& \hfill 2& \hfill -2\end{array}\right][/latex]
Solution
- Add the corresponding entries.
[latex]\begin{array}{l}\hfill \\ A+B=\left[\begin{array}{rrr}\hfill 2& \hfill -10& \hfill -2\\ \hfill 14& \hfill 12& \hfill 10\\ \hfill 4& \hfill -2& \hfill 2\end{array}\right]+\left[\begin{array}{rrr}\hfill 6& \hfill 10& \hfill -2\\ \hfill 0& \hfill -12& \hfill -4\\ \hfill -5& \hfill 2& \hfill -2\end{array}\right]\hfill \\ =\left[\begin{array}{rrr}\hfill 2+6& \hfill -10+10& \hfill -2 - 2\\ \hfill 14+0& \hfill 12 - 12& \hfill 10 - 4\\ \hfill 4 - 5& \hfill -2+2& \hfill 2 - 2\end{array}\right]\hfill \\ =\left[\begin{array}{rrr}\hfill 8& \hfill 0& \hfill -4\\ \hfill 14& \hfill 0& \hfill 6\\ \hfill -1& \hfill 0& \hfill 0\end{array}\right]\hfill \end{array}[/latex]
- Subtract the corresponding entries.
[latex]\begin{array}{l}\hfill \\ A-B=\left[\begin{array}{rrr}\hfill 2& \hfill -10& \hfill -2\\ \hfill 14& \hfill 12& \hfill 10\\ \hfill 4& \hfill -2& \hfill 2\end{array}\right]-\left[\begin{array}{rrr}\hfill 6& \hfill 10& \hfill -2\\ \hfill 0& \hfill -12& \hfill -4\\ \hfill -5& \hfill 2& \hfill -2\end{array}\right]\hfill \\ =\left[\begin{array}{rrr}\hfill 2 - 6& \hfill -10 - 10& \hfill -2+2\\ \hfill 14 - 0& \hfill 12+12& \hfill 10+4\\ \hfill 4+5& \hfill -2 - 2& \hfill 2+2\end{array}\right]\hfill \\ =\left[\begin{array}{rrr}\hfill -4& \hfill -20& \hfill 0\\ \hfill 14& \hfill 24& \hfill 14\\ \hfill 9& \hfill -4& \hfill 4\end{array}\right]\hfill \end{array}[/latex]
Try It 1
Add matrix [latex]A[/latex] and matrix [latex]B[/latex].[latex]A=\left[\begin{array}{rr}\hfill 2& \hfill 6\\ \hfill 1& \hfill 0\\ \hfill 1& \hfill -3\end{array}\right]\text{ and }B=\left[\begin{array}{rr}\hfill 3& \hfill -2\\ \hfill 1& \hfill 5\\ \hfill -4& \hfill 3\end{array}\right][/latex]
Finding Scalar Multiples of a Matrix
Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication. Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school’s current inventory is displayed in the table below.| Lab A | Lab B | |
|---|---|---|
| Computers | 15 | 27 |
| Computer Tables | 16 | 34 |
| Chairs | 16 | 34 |
[latex]{C}_{2013}=\left[\begin{array}{c}15\\ 16\\ 16\end{array}\begin{array}{c}27\\ 34\\ 34\end{array}\right][/latex]
To calculate how much computer equipment will be needed, we multiply all entries in matrix [latex]C[/latex] by 0.15.
[latex]\left(0.15\right){C}_{2013}=\left[\begin{array}{c}\left(0.15\right)15\\ \left(0.15\right)16\\ \left(0.15\right)16\end{array}\begin{array}{c}\left(0.15\right)27\\ \left(0.15\right)34\\ \left(0.15\right)34\end{array}\right]=\left[\begin{array}{c}2.25\\ 2.4\\ 2.4\end{array}\begin{array}{c}4.05\\ 5.1\\ 5.1\end{array}\right][/latex]
We must round up to the next integer, so the amount of new equipment needed is
[latex]\left[\begin{array}{c}3\\ 3\\ 3\end{array}\begin{array}{c}5\\ 6\\ 6\end{array}\right][/latex]
Adding the two matrices as shown below, we see the new inventory amounts.
[latex]\left[\begin{array}{c}15\\ 16\\ 16\end{array}\begin{array}{c}27\\ 34\\ 34\end{array}\right]+\left[\begin{array}{c}3\\ 3\\ 3\end{array}\begin{array}{c}5\\ 6\\ 6\end{array}\right]=\left[\begin{array}{c}18\\ 19\\ 19\end{array}\begin{array}{c}32\\ 40\\ 40\end{array}\right][/latex]
This means
[latex]{C}_{2014}=\left[\begin{array}{c}18\\ 19\\ 19\end{array}\begin{array}{c}32\\ 40\\ 40\end{array}\right][/latex]
Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.
A General Note: Scalar Multiplication
Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given[latex]A=\left[\begin{array}{cccc}{a}_{11}& & & {a}_{12}\\ {a}_{21}& & & {a}_{22}\end{array}\right][/latex]
the scalar multiple [latex]cA[/latex] is
[latex]\begin{array}{l}cA=c\left[\begin{array}{ccc}{a}_{11}& & {a}_{12}\\ {a}_{21}& & {a}_{22}\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{ccc}c{a}_{11}& & c{a}_{12}\\ c{a}_{21}& & c{a}_{22}\end{array}\right]\hfill \end{array}[/latex]
Scalar multiplication is distributive. For the matrices [latex]A,B[/latex], and [latex]C[/latex] with scalars [latex]a[/latex] and [latex]b[/latex],
[latex]\begin{array}{l}\\ \begin{array}{c}a\left(A+B\right)=aA+aB\\ \left(a+b\right)A=aA+bA\end{array}\end{array}[/latex]
Example 6: Multiplying the Matrix by a Scalar
Multiply matrix [latex]A[/latex] by the scalar 3.[latex]A=\left[\begin{array}{cc}8& 1\\ 5& 4\end{array}\right][/latex]
Solution
Multiply each entry in [latex]A[/latex] by the scalar 3.[latex]\begin{array}{l}3A=3\left[\begin{array}{rr}\hfill 8& \hfill 1\\ \hfill 5& \hfill 4\end{array}\right]\hfill \\ = \left[\begin{array}{rr}\hfill 3\cdot 8& \hfill 3\cdot 1\\ \hfill 3\cdot 5& \hfill 3\cdot 4\end{array}\right]\hfill \\ = \left[\begin{array}{rr}\hfill 24& \hfill 3\\ \hfill 15& \hfill 12\end{array}\right]\hfill \end{array}[/latex]
Try It 2
Given matrix [latex]B,\text{}[/latex] find [latex]-2B[/latex] where[latex]B=\left[\begin{array}{cc}4& 1\\ 3& 2\end{array}\right][/latex]
Example 7: Finding the Sum of Scalar Multiples
Find the sum [latex]3A+2B[/latex].[latex]A=\left[\begin{array}{rrr}\hfill 1& \hfill -2& \hfill 0\\ \hfill 0& \hfill -1& \hfill 2\\ \hfill 4& \hfill 3& \hfill -6\end{array}\right]\text{ and }B=\left[\begin{array}{rrr}\hfill -1& \hfill 2& \hfill 1\\ \hfill 0& \hfill -3& \hfill 2\\ \hfill 0& \hfill 1& \hfill -4\end{array}\right][/latex]
Solution
First, find [latex]3A,\text{}[/latex] then [latex]2B[/latex].[latex]\begin{array}{l}\begin{array}{l}\hfill \\ \hfill \\ 3A=\left[\begin{array}{lll}3\cdot 1\hfill & 3\left(-2\right)\hfill & 3\cdot 0\hfill \\ 3\cdot 0\hfill & 3\left(-1\right)\hfill & 3\cdot 2\hfill \\ 3\cdot 4\hfill & 3\cdot 3\hfill & 3\left(-6\right)\hfill \end{array}\right]\hfill \end{array}\hfill \\ =\left[\begin{array}{rrr}\hfill 3& \hfill -6& \hfill 0\\ \hfill 0& \hfill -3& \hfill 6\\ \hfill 12& \hfill 9& \hfill -18\end{array}\right]\hfill \end{array}[/latex]
[latex]\begin{array}{l}\begin{array}{l}\hfill \\ \hfill \\ 2B=\left[\begin{array}{lll}2\left(-1\right)\hfill & 2\cdot 2\hfill & 2\cdot 1\hfill \\ 2\cdot 0\hfill & 2\left(-3\right)\hfill & 2\cdot 2\hfill \\ 2\cdot 0\hfill & 2\cdot 1\hfill & 2\left(-4\right)\hfill \end{array}\right]\hfill \end{array}\hfill \\ =\left[\begin{array}{rrr}\hfill -2& \hfill 4& \hfill 2\\ \hfill 0& \hfill -6& \hfill 4\\ \hfill 0& \hfill 2& \hfill -8\end{array}\right]\hfill \end{array}[/latex]
Now, add [latex]3A+2B[/latex].
[latex]\begin{array}{l}\hfill \\ \hfill \\ 3A+2B=\left[\begin{array}{rrr}\hfill 3& \hfill -6& \hfill 0\\ \hfill 0& \hfill -3& \hfill 6\\ \hfill 12& \hfill 9& \hfill -18\end{array}\right]+\left[\begin{array}{rrr}\hfill -2& \hfill 4& \hfill 2\\ \hfill 0& \hfill -6& \hfill 4\\ \hfill 0& \hfill 2& \hfill -8\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{rrr}\hfill 3 - 2& \hfill -6+4& \hfill 0+2\\ \hfill 0+0& \hfill -3 - 6& \hfill 6+4\\ \hfill 12+0& \hfill 9+2& \hfill -18 - 8\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{rrr}\hfill 1& \hfill -2& \hfill 2\\ \hfill 0& \hfill -9& \hfill 10\\ \hfill 12& \hfill 11& \hfill -26\end{array}\right]\hfill \end{array}[/latex]
Finding the Product of Two Matrices
In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If [latex]A[/latex] is an [latex]\text{ }m\text{ }\times \text{ }r\text{ }[/latex] matrix and [latex]B[/latex] is an [latex]\text{ }r\text{ }\times \text{ }n\text{ }[/latex] matrix, then the product matrix [latex]AB[/latex] is an [latex]\text{ }m\text{ }\times \text{ }n\text{ }[/latex] matrix. For example, the product [latex]AB[/latex] is possible because the number of columns in [latex]A[/latex] is the same as the number of rows in [latex]B[/latex]. If the inner dimensions do not match, the product is not defined.
Figure 1[latex]A=\left[\begin{array}{rrr}\hfill {a}_{11}& \hfill {a}_{12}& \hfill {a}_{13}\\ \hfill {a}_{21}& \hfill {a}_{22}& \hfill {a}_{23}\end{array}\right]\text{ and }B=\left[\begin{array}{rrr}\hfill {b}_{11}& \hfill {b}_{12}& \hfill {b}_{13}\\ \hfill {b}_{21}& \hfill {b}_{22}& \hfill {b}_{23}\\ \hfill {b}_{31}& \hfill {b}_{32}& \hfill {b}_{33}\end{array}\right][/latex]
Multiply and add as follows to obtain the first entry of the product matrix [latex]AB[/latex].
- To obtain the entry in row 1, column 1 of [latex]AB,\text{}[/latex] multiply the first row in [latex]A[/latex] by the first column in [latex]B[/latex], and add.
[latex]\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]\cdot \left[\begin{array}{c}{b}_{11}\\ {b}_{21}\\ {b}_{31}\end{array}\right]={a}_{11}\cdot {b}_{11}+{a}_{12}\cdot {b}_{21}+{a}_{13}\cdot {b}_{31}[/latex]
- To obtain the entry in row 1, column 2 of [latex]AB,\text{}[/latex] multiply the first row of [latex]A[/latex] by the second column in [latex]B[/latex], and add.
[latex]\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]\cdot \left[\begin{array}{c}{b}_{12}\\ {b}_{22}\\ {b}_{32}\end{array}\right]={a}_{11}\cdot {b}_{12}+{a}_{12}\cdot {b}_{22}+{a}_{13}\cdot {b}_{32}[/latex]
- To obtain the entry in row 1, column 3 of [latex]AB,\text{}[/latex] multiply the first row of [latex]A[/latex] by the third column in [latex]B[/latex], and add.
[latex]\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]\cdot \left[\begin{array}{c}{b}_{13}\\ {b}_{23}\\ {b}_{33}\end{array}\right]={a}_{11}\cdot {b}_{13}+{a}_{12}\cdot {b}_{23}+{a}_{13}\cdot {b}_{33}[/latex]
[latex]AB=\left[\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{11}+{a}_{12}\cdot {b}_{21}+{a}_{13}\cdot {b}_{31}\\ \end{array}\\ {a}_{21}\cdot {b}_{11}+{a}_{22}\cdot {b}_{21}+{a}_{23}\cdot {b}_{31}\end{array}\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{12}+{a}_{12}\cdot {b}_{22}+{a}_{13}\cdot {b}_{32}\\ \end{array}\\ {a}_{21}\cdot {b}_{12}+{a}_{22}\cdot {b}_{22}+{a}_{23}\cdot {b}_{32}\end{array}\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{13}+{a}_{12}\cdot {b}_{23}+{a}_{13}\cdot {b}_{33}\\ \end{array}\\ {a}_{21}\cdot {b}_{13}+{a}_{22}\cdot {b}_{23}+{a}_{23}\cdot {b}_{33}\end{array}\right][/latex]
A General Note: Properties of Matrix Multiplication
For the matrices [latex]A,B,\text{}[/latex] and [latex]C[/latex] the following properties hold.- Matrix multiplication is associative: [latex]\left(AB\right)C=A\left(BC\right)[/latex].
- Matrix multiplication is distributive: [latex]\begin{array}{l}\begin{array}{l}\\ C\left(A+B\right)=CA+CB,\end{array}\hfill \\ \left(A+B\right)C=AC+BC.\hfill \end{array}[/latex]
Example 8: Multiplying Two Matrices
Multiply matrix [latex]A[/latex] and matrix [latex]B[/latex].[latex]A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\text{ and }B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right][/latex]
Solution
First, we check the dimensions of the matrices. Matrix [latex]A[/latex] has dimensions [latex]2\times 2[/latex] and matrix [latex]B[/latex] has dimensions [latex]2\times 2[/latex]. The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions [latex]2\times 2[/latex]. We perform the operations outlined previously.
Figure 2Example 9: Multiplying Two Matrices
Given [latex]A[/latex] and [latex]B:[/latex]- Find [latex]AB[/latex].
- Find [latex]BA[/latex].
[latex]A=\left[\begin{array}{l}\begin{array}{ccc}-1& 2& 3\end{array}\hfill \\ \begin{array}{ccc}4& 0& 5\end{array}\hfill \end{array}\right]\text{ and }B=\left[\begin{array}{c}5\\ -4\\ 2\end{array}\begin{array}{c}-1\\ 0\\ 3\end{array}\right][/latex]
Solution
- As the dimensions of [latex]A[/latex] are [latex]2\text{}\times \text{}3[/latex] and the dimensions of [latex]B[/latex] are [latex]3\text{}\times \text{}2,\text{}[/latex] these matrices can be multiplied together because the number of columns in [latex]A[/latex] matches the number of rows in [latex]B[/latex]. The resulting product will be a [latex]2\text{}\times \text{}2[/latex] matrix, the number of rows in [latex]A[/latex] by the number of columns in [latex]B[/latex].
[latex]\begin{array}{l}\hfill \\ AB=\left[\begin{array}{rrr}\hfill -1& \hfill 2& \hfill 3\\ \hfill 4& \hfill 0& \hfill 5\end{array}\right]\text{ }\left[\begin{array}{rr}\hfill 5& \hfill -1\\ \hfill -4& \hfill 0\\ \hfill 2& \hfill 3\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{rr}\hfill -1\left(5\right)+2\left(-4\right)+3\left(2\right)& \hfill -1\left(-1\right)+2\left(0\right)+3\left(3\right)\\ \hfill 4\left(5\right)+0\left(-4\right)+5\left(2\right)& \hfill 4\left(-1\right)+0\left(0\right)+5\left(3\right)\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{rr}\hfill -7& \hfill 10\\ \hfill 30& \hfill 11\end{array}\right]\hfill \end{array}[/latex]
- The dimensions of [latex]B[/latex] are [latex]3\times 2[/latex] and the dimensions of [latex]A[/latex] are [latex]2\times 3[/latex]. The inner dimensions match so the product is defined and will be a [latex]3\times 3[/latex] matrix.
[latex]\begin{array}{l}\hfill \\ BA=\left[\begin{array}{rr}\hfill 5& \hfill -1\\ \hfill -4& \hfill 0\\ \hfill 2& \hfill 3\end{array}\right]\text{ }\left[\begin{array}{rrr}\hfill -1& \hfill 2& \hfill 3\\ \hfill 4& \hfill 0& \hfill 5\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{rrr}\hfill 5\left(-1\right)+-1\left(4\right)& \hfill 5\left(2\right)+-1\left(0\right)& \hfill 5\left(3\right)+-1\left(5\right)\\ \hfill -4\left(-1\right)+0\left(4\right)& \hfill -4\left(2\right)+0\left(0\right)& \hfill -4\left(3\right)+0\left(5\right)\\ \hfill 2\left(-1\right)+3\left(4\right)& \hfill 2\left(2\right)+3\left(0\right)& \hfill 2\left(3\right)+3\left(5\right)\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{rrr}\hfill -9& \hfill 10& \hfill 10\\ \hfill 4& \hfill -8& \hfill -12\\ \hfill 10& \hfill 4& \hfill 21\end{array}\right]\hfill \end{array}[/latex]
Analysis of the Solution
Notice that the products [latex]AB[/latex] and [latex]BA[/latex] are not equal.[latex]AB=\left[\begin{array}{cc}-7& 10\\ 30& 11\end{array}\right]\ne \left[\begin{array}{ccc}-9& 10& 10\\ 4& -8& -12\\ 10& 4& 21\end{array}\right]=BA[/latex]
This illustrates the fact that matrix multiplication is not commutative.
Q & A
Is it possible for AB to be defined but not BA?
Yes, consider a matrix A with dimension [latex]3\times 4[/latex] and matrix B with dimension [latex]4\times 2[/latex]. For the product AB the inner dimensions are 4 and the product is defined, but for the product BA the inner dimensions are 2 and 3 so the product is undefined.Example 10: Using Matrices in Real-World Problems
Let’s return to the problem presented at the opening of this section. We have the table below, representing the equipment needs of two soccer teams.| Wildcats | Mud Cats | |
|---|---|---|
| Goals | 6 | 10 |
| Balls | 30 | 24 |
| Jerseys | 14 | 20 |
| Goal | $300 |
| Ball | $10 |
| Jersey | $30 |
[latex]E=\left[\begin{array}{c}6\\ 30\\ 14\end{array}\begin{array}{c}10\\ 24\\ 20\end{array}\right][/latex]
The cost matrix is written as
[latex]C=\left[\begin{array}{ccc}300& 10& 30\end{array}\right][/latex]
We perform matrix multiplication to obtain costs for the equipment.
[latex]\begin{array}{l}\hfill \\ \hfill \\ CE=\left[\begin{array}{rrr}\hfill 300& \hfill 10& \hfill 30\end{array}\right]\cdot \left[\begin{array}{rr}\hfill 6& \hfill 10\\ \hfill 30& \hfill 24\\ \hfill 14& \hfill 20\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{rr}\hfill 300\left(6\right)+10\left(30\right)+30\left(14\right)& \hfill 300\left(10\right)+10\left(24\right)+30\left(20\right)\end{array}\right]\hfill \\ \text{ }=\left[\begin{array}{rr}\hfill 2,520& \hfill 3,840\end{array}\right]\hfill \end{array}[/latex]
The total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840.
How To: Given a matrix operation, evaluate using a calculator.
- Save each matrix as a matrix variable [latex]\left[A\right],\left[B\right],\left[C\right],..[/latex].
- Enter the operation into the calculator, calling up each matrix variable as needed.
- If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will display an error message.
Example 11: Using a Calculator to Perform Matrix Operations
Find [latex]AB-C[/latex] given[latex]A=\left[\begin{array}{rrr}\hfill -15& \hfill 25& \hfill 32\\ \hfill 41& \hfill -7& \hfill -28\\ \hfill 10& \hfill 34& \hfill -2\end{array}\right],B=\left[\begin{array}{rrr}\hfill 45& \hfill 21& \hfill -37\\ \hfill -24& \hfill 52& \hfill 19\\ \hfill 6& \hfill -48& \hfill -31\end{array}\right],\text{and }C=\left[\begin{array}{rrr}\hfill -100& \hfill -89& \hfill -98\\ \hfill 25& \hfill -56& \hfill 74\\ \hfill -67& \hfill 42& \hfill -75\end{array}\right][/latex].
Solution
On the matrix page of the calculator, we enter matrix [latex]A[/latex] above as the matrix variable [latex]\left[A\right][/latex], matrix [latex]B[/latex] above as the matrix variable [latex]\left[B\right][/latex], and matrix [latex]C[/latex] above as the matrix variable [latex]\left[C\right][/latex]. On the home screen of the calculator, we type in the problem and call up each matrix variable as needed.[latex]\left[A\right]\times \left[B\right]-\left[C\right][/latex]
The calculator gives us the following matrix.
[latex]\left[\begin{array}{rrr}\hfill -983& \hfill -462& \hfill 136\\ \hfill 1,820& \hfill 1,897& \hfill -856\\ \hfill -311& \hfill 2,032& \hfill 413\end{array}\right][/latex]
Key Concepts
- A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.
- The dimensions of a matrix refer to the number of rows and the number of columns. A [latex]3\times 2[/latex] matrix has three rows and two columns.
- We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix.
- Scalar multiplication involves multiplying each entry in a matrix by a constant.
- Scalar multiplication is often required before addition or subtraction can occur.
- Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must match the number of rows in the second.
- The product of two matrices, [latex]A[/latex] and [latex]B[/latex], is obtained by multiplying each entry in row 1 of [latex]A[/latex] by each entry in column 1 of [latex]B[/latex]; then multiply each entry of row 1 of [latex]A[/latex] by each entry in columns 2 of [latex]B,\text{}[/latex] and so on.
- Many real-world problems can often be solved using matrices.
- We can use a calculator to perform matrix operations after saving each matrix as a matrix variable.
Glossary
- column
- a set of numbers aligned vertically in a matrix
- entry
- an element, coefficient, or constant in a matrix
- matrix
- a rectangular array of numbers
- row
- a set of numbers aligned horizontally in a matrix
- scalar multiple
- an entry of a matrix that has been multiplied by a scalar
Section Exercises
1. Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together. 2. Can we multiply any column matrix by any row matrix? Explain why or why not. 3. Can both the products [latex]AB[/latex] and [latex]BA[/latex] be defined? If so, explain how; if not, explain why. 4. Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together. 5. Does matrix multiplication commute? That is, does [latex]AB=BA?[/latex] If so, prove why it does. If not, explain why it does not. For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.[latex]A=\left[\begin{array}{cc}1& 3\\ 0& 7\end{array}\right],B=\left[\begin{array}{cc}2& 14\\ 22& 6\end{array}\right],C=\left[\begin{array}{cc}1& 5\\ 8& 92\\ 12& 6\end{array}\right],D=\left[\begin{array}{cc}10& 14\\ 7& 2\\ 5& 61\end{array}\right],E=\left[\begin{array}{cc}6& 12\\ 14& 5\end{array}\right],F=\left[\begin{array}{cc}0& 9\\ 78& 17\\ 15& 4\end{array}\right][/latex]
6. [latex]A+B[/latex] 7. [latex]C+D[/latex] 8. [latex]A+C[/latex] 9. [latex]B-E[/latex] 10. [latex]C+F[/latex] 11. [latex]D-B[/latex] For the following exercises, use the matrices below to perform scalar multiplication.[latex]A=\left[\begin{array}{rr}\hfill 4& \hfill 6\\ \hfill 13& \hfill 12\end{array}\right],B=\left[\begin{array}{rr}\hfill 3& \hfill 9\\ \hfill 21& \hfill 12\\ \hfill 0& \hfill 64\end{array}\right],C=\left[\begin{array}{rrrr}\hfill 16& \hfill 3& \hfill 7& \hfill 18\\ \hfill 90& \hfill 5& \hfill 3& \hfill 29\end{array}\right],D=\left[\begin{array}{rrr}\hfill 18& \hfill 12& \hfill 13\\ \hfill 8& \hfill 14& \hfill 6\\ \hfill 7& \hfill 4& \hfill 21\end{array}\right][/latex]
12. [latex]5A[/latex] 13. [latex]3B[/latex] 14. [latex]-2B[/latex] 15. [latex]-4C[/latex] 16. [latex]\frac{1}{2}C[/latex] 17. [latex]100D[/latex] For the following exercises, use the matrices below to perform matrix multiplication.[latex]A=\left[\begin{array}{rr}\hfill -1& \hfill 5\\ \hfill 3& \hfill 2\end{array}\right],B=\left[\begin{array}{rrr}\hfill 3& \hfill 6& \hfill 4\\ \hfill -8& \hfill 0& \hfill 12\end{array}\right],C=\left[\begin{array}{rr}\hfill 4& \hfill 10\\ \hfill -2& \hfill 6\\ \hfill 5& \hfill 9\end{array}\right],D=\left[\begin{array}{rrr}\hfill 2& \hfill -3& \hfill 12\\ \hfill 9& \hfill 3& \hfill 1\\ \hfill 0& \hfill 8& \hfill -10\end{array}\right][/latex]
18. [latex]AB[/latex] 19. [latex]BC[/latex] 20. [latex]CA[/latex] 21. [latex]BD[/latex] 22. [latex]DC[/latex] 23. [latex]CB[/latex] For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.[latex]A=\left[\begin{array}{rr}\hfill 2& \hfill -5\\ \hfill 6& \hfill 7\end{array}\right],B=\left[\begin{array}{rr}\hfill -9& \hfill 6\\ \hfill -4& \hfill 2\end{array}\right],C=\left[\begin{array}{rr}\hfill 0& \hfill 9\\ \hfill 7& \hfill 1\end{array}\right],D=\left[\begin{array}{rrr}\hfill -8& \hfill 7& \hfill -5\\ \hfill 4& \hfill 3& \hfill 2\\ \hfill 0& \hfill 9& \hfill 2\end{array}\right],E=\left[\begin{array}{rrr}\hfill 4& \hfill 5& \hfill 3\\ \hfill 7& \hfill -6& \hfill -5\\ \hfill 1& \hfill 0& \hfill 9\end{array}\right][/latex]
24. [latex]A+B-C[/latex] 25. [latex]4A+5D[/latex] 26. [latex]2C+B[/latex] 27. [latex]3D+4E[/latex] 28. [latex]C - 0.5D[/latex] 29. [latex]100D - 10E[/latex] For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: [latex]{A}^{2}=A\cdot A[/latex])[latex]A=\left[\begin{array}{rr}\hfill -10& \hfill 20\\ \hfill 5& \hfill 25\end{array}\right],B=\left[\begin{array}{rr}\hfill 40& \hfill 10\\ \hfill -20& \hfill 30\end{array}\right],C=\left[\begin{array}{rr}\hfill -1& \hfill 0\\ \hfill 0& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right][/latex]
30. [latex]AB[/latex] 31. [latex]BA[/latex] 32. [latex]CA[/latex] 33. [latex]BC[/latex] 34. [latex]{A}^{2}[/latex] 35. [latex]{B}^{2}[/latex] 36. [latex]{C}^{2}[/latex] 37. [latex]{B}^{2}{A}^{2}[/latex] 38. [latex]{A}^{2}{B}^{2}[/latex] 39. [latex]{\left(AB\right)}^{2}[/latex] 40. [latex]{\left(BA\right)}^{2}[/latex] For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: [latex]{A}^{2}=A\cdot A[/latex])[latex]A=\left[\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 2& \hfill 3\end{array}\right],B=\left[\begin{array}{rrr}\hfill -2& \hfill 3& \hfill 4\\ \hfill -1& \hfill 1& \hfill -5\end{array}\right],C=\left[\begin{array}{rr}\hfill 0.5& \hfill 0.1\\ \hfill 1& \hfill 0.2\\ \hfill -0.5& \hfill 0.3\end{array}\right],D=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill -1\\ \hfill -6& \hfill 7& \hfill 5\\ \hfill 4& \hfill 2& \hfill 1\end{array}\right][/latex]
41. [latex]AB[/latex] 42. [latex]BA[/latex] 43. [latex]BD[/latex] 44. [latex]DC[/latex] 45. [latex]{D}^{2}[/latex] 46. [latex]{A}^{2}[/latex] 47. [latex]{D}^{3}[/latex] 48. [latex]\left(AB\right)C[/latex] 49. [latex]A\left(BC\right)[/latex] For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.[latex]A=\left[\begin{array}{rrr}\hfill -2& \hfill 0& \hfill 9\\ \hfill 1& \hfill 8& \hfill -3\\ \hfill 0.5& \hfill 4& \hfill 5\end{array}\right],B=\left[\begin{array}{rrr}\hfill 0.5& \hfill 3& \hfill 0\\ \hfill -4& \hfill 1& \hfill 6\\ \hfill 8& \hfill 7& \hfill 2\end{array}\right],C=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 1& \hfill 0& \hfill 1\end{array}\right][/latex]
50. [latex]AB[/latex] 51. [latex]BA[/latex] 52. [latex]CA[/latex] 53. [latex]BC[/latex] 54. [latex]ABC[/latex] For the following exercises, use the matrix below to perform the indicated operation on the given matrix.[latex]B=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 0\end{array}\right][/latex]
55. [latex]{B}^{2}[/latex] 56. [latex]{B}^{3}[/latex] 57. [latex]{B}^{4}[/latex] 58. [latex]{B}^{5}[/latex] 59. Using the above questions, find a formula for [latex]{B}^{n}[/latex]. Test the formula for [latex]{B}^{201}[/latex] and [latex]{B}^{202},\text{}[/latex] using a calculator.Licenses & Attributions
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