Reading: Matrices and Matrix Operations

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. Table 1 shows the needs of both teams.

Table 1
	Wildcats	Mud Cats
Goals	6	10
Balls	30	24
Jerseys	14	20

A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.

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 and are shown below. [latex-display]A=\matrix{{1}&{2}\\{3}&{4}}[/latex-display] [latex-display]\displaystyle{B = \left[\matrix{{1}&{2}&{7}\\{0}&{5}&{6}\\{7}&{8}&{2}}\right][/latex-display] [latex-display]\displaystyle{C = \left[\matrix{{1}&{3}\\{0}&{2}\\{3} & {1}}\right][/latex-display]

Describing Matrices

A matrix is often referred to by its size or dimensions: m × n indicating m rows and n columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix A identified as a_ij we look for the entry in row_i column_j.  In the matrix shown below, the entry in row 2, column 3 is a_{23 = [latex]\displaystyle{A}={\left[\matrix{{a}_{{11}}&{a}_{{12}}&{a}_{{13}}\\{a}_{{21}}&{a}_{{22}}&{a}_{{23}}\\{a}_{{31}}&{a}_{{32}}&{a}_{{33}}}\right][/latex]} A square matrix is a matrix with dimensions meaning that it has the same number of rows as columns. The matrix above is an example of a square matrix. A row matrix is a matrix consisting of one row with dimensions 1 × n [latex-display]\displaystyle{\left[\matrix{{a}_{11}&{a}_{{21}}&{a}_{{31}}\dots&{a}_{{n1}}\right][/latex-display] A column matrix is a matrix consisting of one column with dimensions m × 1 [latex-display]\displaystyle{\left[\matrix{{a}_{11}\\{a}_{{21}}\\{a}_{{31}}\right][/latex-display]   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.

Matrices

A matrix is a rectangular array of numbers that is usually named by a capital letter: A, B, C, and so on. Each entry in a matrix is referred to as a_ij, such that represents the row and represents the column. Matrices are often referred to by their dimensions: m × n indicating m rows and n columns.

Example 1

Given matrix find the dimensions of the given matrix and locating entries: [latex-display]\displaystyle{A}={\left[\matrix{{2}&{1}&{0}\\{2}&{4}&{7}\\{3}&{1}&-{2}}\right]}[/latex-display]

1. What are the dimensions of matrix A
2. What are the entries at and a₃₁ and a₂₂

Solutions

1. The dimensions are 3 × 3 because there are three rows and three columns.
2. Entry a₃₁ is the number at row 3, column 1, which is 3. The entry a₂₂ 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 3 × 3 matrix and another 3 × 3 matrix, but we cannot add or subtract a 2 × 3 matrix and a 3 × 3 matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

A General Note

Given matrices A and B of like dimensions, addition and subtraction of A and B will produce matrix C or matrix D of the same dimension. A + B = C such that a_ij + b_ij = c_ij A − B = D such that a_ij− b_ij= d_ij Matrix addition is commutative. A + B = B + A It is also associative. (A + B) + C = A + (B + C)

Example 2

1. Find the sum of A and B given[latex]\displaystyle{A}={\left[\matrix{{a}&{b}\\{c}&{d}}\right]{\text{and}} {B}={\left[\matrix{{e}&{f}\\{g}&{h}}\right][/latex]
2. Find the sum of A and B, given[latex]\displaystyle{A}={\left[\matrix{{4}&{1}\\{3}&{2}}\right] {\text{and}} {B}={\left[\matrix{{5}&{9}\\{0}&{7}}\right][/latex]
3. Find the difference of A and B [latex]\displaystyle{A}={\left[\matrix{-{2}&{3}\\{0}&{1}}\right]{\text{and}} {B}={\left[\matrix{{8}&{1}\\{5}&{4}}\right]}[/latex]
4. Given A and B: [latex]\displaystyle{A}={\left[\matrix{{2}&-{10}&-{2}\\{14}&{12}&{10}\\{4}&-{2}&{2}}\right]} {\text{and}} {B}={\left[\matrix{{6}&{10}&-{2}\\{0}&-{12}&-{4}\\-{5}&{2}&-{2}}\right]}[/latex]

1. Find the sum.
2. Find the difference.

• Add matrix A and matrix B [latex]\displaystyle{A}={\left[\matrix{{2}&{6}\\{1}&{0}\\{1}&-{3}}\right]}{\text{and}}{B}={\left[\matrix{{3}&-{2}\\{1}&{5}\\-{4}&{3}}\right]}[/latex]

Solutions

1. [latex]\displaystyle{A}+{B}={\left[\matrix{{a}&{b}\\{c}&{d}}\right]}+{\left[\matrix{{e}&{f}\\{g}&{h}}\right]}={\left[\matrix{{a}+{e}&{b}+{f}\\{c}+{g}&{d}+{h}}\right]}[/latex]
2. Add corresponding entries. Add the entry in row 1, column 1, a₁₁, of matrix A to the entry in row 1, column 1, b₁₁, of B. Continue the pattern until all entries have been added. [latex]\displaystyle{A}+{B}={\left[\matrix{{4}&{1}\\{3}&{2}}\right]}+{\left[\matrix{{5}&{9}\\{0}&{7}}\right]}={\left[\matrix{{4}+{5}&{1}+{9}\\{3}+{0}&{2}+{7}}\right]}={\left[\matrix{{9}&{10}\\{3}&{9}}]\right}[/latex]
3. We subtract the corresponding entries of each matrix. [latex]\displaystyle{A}-{B}={\left[\matrix{-{2}&{3}\\{0}&{1}}\right]}-{\left[\matrix{{8}&{1}\\{5}&{4}}\right]}={\left[\matrix{-{2}-{8}&{3}-{1}\\{0}-{5}&{1}-{4}}\right]}={\left[\matrix{-{10}&{2}\\-{5}&-{3}}\right]}[/latex]
4. 1. Add the corresponding entries [latex]\displaystyle{A}+{B}={\left[\matrix{{2}&-{10}&-{2}\\{14}&{12}&{10}\\{4}&-{2}&{2}}\right]}+{\left[\matrix{{6}&{10}&-{2}\\{0}&-{12}&-{4}\\-{5}&{2}&-{2}}\right]}[/latex]
   2. Subtract the corresponding entries [latex]\displaystyle{A}-{B}={\left[\matrix{{2}&-{10}&-{2}\\{14}&{12}&{10}\\{4}&-{2}&{2}}\right]}-{\left[\matrix{{6}&{10}&-{2}\\{0}&-{12}&-{4}\\-{5}&{2}&-{2}}\right]}[/latex]
5. [latex]\displaystyle{A}+{B}={\left[\matrix{{2}&{6}\\{1}&{0}\\{1}&-{3}}\right]}+{\left[\matrix{{3}&-{2}\\{1}&{5}\\-{4}&{3}}\right]}={\left[\matrix{{2}+{3}&{6}+{(-{2})}\\{1}+{1}&{0}+{5}\\{1}+{(-{4})}&-{3}+{3}}\right]}={\left[\matrix{{5}&{4}\\{2}&{5}\\-{3}&{0}}\right]}[/latex]

Finding Scalar Multiples of a Matrix

Scalar Multiplication

Example 3

1. Multiply matrix A by the scalar 3.

2. Given matrix B find –2B where

3.  Find the sum 3A + 2B

Solutions

1. [latex]\displaystyle{3}{A}={3}{\left[\matrix{{8}&{1}\\{5}&{4}}\right]}={\left[\matrix{{3}\cdot{8}&{3}\cdot{1}\\{3}\cdot{5}&{3}\cdot{4}}\\right]}={\left[\matrix{{24}&{3}\\{15}&{12}}\right]}[/latex]
2. [latex]\displaystyle-{2}{B}={\left[\matrix{-{8}&-{2}\\-{6}&-{4}}\right]}[/latex]
3. First, find 3A then 2B [latex]\displaystyle{3}{A}={\left[\matrix{{3}\cdot{1}&{3}{(-{2})}&{3}\cdot{0}\\{3}\cdot{0}&{3}{(-{1})}&{3}\cdot{2}\\{3}\cdot{4}&{3}\cdot{3}&{3}{(-{6})}}\right]}={\left[\matrix{{3}&-{6}&{0}\\{0}&-{3}&{6}\\{12}&{9}&-{18}}\right]}[/latex].  Now, add 3A + 2B.[latex]\displaystyle{3}{A}+{2}{B}={\left[\matrix{{3}&-{6}&{0}\\{0}&-{3}&{6}\\{12}&{9}&-{18}}\right]}+{\left[\matrix{-{2}&{4}&{2}\\{0}&-{6}&{4}\\{0}&{2}&-{8}}\right]}[/latex]

Finding the Product of Two Matrices

1. To obtain the entry in row 1, column 1 of AB, multiply the first row in A by the first column in B, and add [latex]\displaystyle{[{a}_{{11}} {a}_{{12}} {a}_{{13}}]}\cdot{\left[\matrix{{b}_{{11}}\\{b}_{{21}}\\{b}_{{31}}}\right]={a}_{{11}}\cdot{b}_{{11}}+{a}_{{12}}\cdot{b}_{{21}}+{a}_{{13}}\cdot{b}_{{31}}[/latex]
2. To obtain the entry in row 1, column 2 of AB, multiply the second row in A by the second column in B, and add [latex]\displaystyle{[{a}_{{11}} {a}_{{12}} {a}_{{13}}]}\cdot{\left[\matrix{{b}_{{12}}\\{b}_{{22}}\\{b}_{{32}}}\right]={a}_{{11}}\cdot{b}_{{12}}+{a}_{{12}}\cdot{b}_{{22}}+{a}_{{13}}\cdot{b}_{{32}}[/latex]
3. To obtain the entry in row 1, column 3 of AB, multiply the third row in A by the third column in B, and add [latex]\displaystyle{[{a}_{{11}} {a}_{{12}} {a}_{{13}}]}\cdot{\left[\matrix{{b}_{{13}}\\{b}_{{23}}\\{b}_{{33}}}\right]={a}_{{11}}\cdot{b}_{{13}}+{a}_{{12}}\cdot{b}_{{23}}+{a}_{{13}}\cdot{b}_{{33}}[/latex]

Properties of Matrix Multiplication

• Matrix multiplication is associative: (AB)C=A(BC).
• Matrix multiplication is distributive: C(A+B)=CA+CB,(A+B)C=AC+BC

Example 4

1. Multiply matrix A and matrix B. [latex]\displaystyle{A}={\left[\matrix{{1}&{2}\\{3}&{4}}\right]}{\text{and}}{B}={\left[\matrix{{5}&{6}\\{7}&{8}}\right]}[/latex]
2. Given A and B: [latex-display]\displaystyle{A}={\left[\matrix{-{1}&{2}&{3}\\{4}&{0}&{5}}\right]}{\text{and}}{B}={\left[\matrix{{5}&-{1}\\-{4}&{0}\\{2}&{3}}\right]}[/latex-display]
   1. Find AB
   2. Find BA

Solutions

1. First, we check the dimensions of the matrices. Matrix A has dimensions 2 × 2 and matrix B has dimensions 2 × 2. The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions 2 × 2. We perform the operations outlined previously.[latex]\displaystyle{A}{B}={\left[\matrix{{1}&{2}\\{3}&{4}}\right]}\cdot{\left[\matrix{{5}&{6}\\{7}&{8}}\right]}={\left[\matrix{{1}{({5})}+{2}{({7})}&{1}{({6})}+{2}{({8})}\\{3}{({5})}+{4}{({7})}&{3}{({6})}+{4}{({8})}}\right]}={\left[\matrix{{19}&{22}\\{43}&{50}}\right]}[/latex]
2. Notice that the products AB and BA are not equal.[latex]\displaystyle{A}{B}={\left[\matrix{-{7}&{10}\\{30}&{11}}\right]}\neq{\left[\matrix{-{9}&{10}&{10}\\{4}&-{8}&-{12}\\{10}&{4}&{21}}\right]}={B}{A}[/latex]This illustrates the fact that matrix multiplication is not commutative.

Q&A

Example 5

How To

1. Save each matrix as a matrix variable
2. Enter the operation into the calculator, calling up each matrix variable as needed.
3. If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will display an error message.

Example 6

Solutions

Media

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 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, and is obtained by multiplying each entry in row 1 of by each entry in column 1 of then multiply each entry of row 1 of by each entry in columns 2 of 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.