Solving Systems with Cramer's Rule

Using Cramer’s Rule to Solve a System of Two Equations in Two Variables

Evaluating the Determinant of a 2×2 Matrix

A General Note: Find the Determinant of a 2 × 2 Matrix

The determinant of a [latex]2\text{ }\times \text{ }2[/latex] matrix, given

[latex]A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right][/latex]

is defined as

Figure 1

Notice the change in notation. There are several ways to indicate the determinant, including [latex]\mathrm{det}\left(A\right)[/latex] and replacing the brackets in a matrix with straight lines, [latex]|A|[/latex].

Using Cramer’s Rule to Solve a System of Two Equations in Two Variables

[latex]\begin{array}\text{ }b_{2}a_{1}x+b_{2}b_{1}y=b_{2}c_{1} \hfill& \text{Multiply }R_{1}\text{ by }b_{2} \\−b_{1}a_{2}x−b_{1}b_{2}y=−b_{1}c_{2} \hfill& \text{Multiply }R_{2}\text{ by }−b_{2} \\ \text{______________________} \\ b_{2}a_{1}x−b_{1}a_{2}x=−b_{2}c_{1}−b_{1}c_{2}\end{array}[/latex]

• [latex]D:[/latex] determinant of the coefficient matrix
• [latex]{D}_{x}:[/latex] determinant of the numerator in the solution of [latex]x[/latex]
  [latex]x=\frac{{D}_{x}}{D}[/latex]
• [latex]{D}_{y}:[/latex] determinant of the numerator in the solution of [latex]y[/latex]
  [latex]y=\frac{{D}_{y}}{D}[/latex]

 Using Cramer’s Rule to Solve a System of Three Equations in Three Variables

Evaluating the Determinant of a 3 × 3 Matrix

1. Augment [latex]A[/latex] with the first two columns.
   [latex]\mathrm{det}\left(A\right)=|\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\ {a}_{2}& {b}_{2}& {c}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}\end{array}|\begin{array}{c}{a}_{1}\\ {a}_{2}\\ {a}_{3}\end{array}\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\end{array}|[/latex]
2. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.
3. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.

Figure 2

Using Cramer’s Rule to Solve a System of Three Equations in Three Variables

Figure 3

Figure 4

Solution

We begin by finding the determinants [latex]D,{D}_{x},\text{and }{D}_{y}[/latex].

[latex]D=|\begin{array}{cc}3& -2\\ 6& -4\end{array}|=3\left(-4\right)-6\left(-2\right)=0[/latex]

We know that a determinant of zero means that either the system has no solution or it has an infinite number of solutions. To see which one, we use the process of elimination. Our goal is to eliminate one of the variables.

1. Multiply equation (1) by [latex]-2[/latex].
2. Add the result to equation [latex]\left(2\right)[/latex].

[latex]\begin{array}\text{ }\hfill−6x+4y=−8 \\ \hfill6x−4y=0 \\ \hfill\text{_____________} \\ \hfill 0=8\end{array}[/latex]

We obtain the equation [latex]0=-8[/latex], which is false. Therefore, the system has no solution. Graphing the system reveals two parallel lines.

Figure 5

Solution

Let’s find the determinant first. Set up a matrix augmented by the first two columns.

[latex]|\begin{array}{rrr}\hfill 1& \hfill -2& \hfill 3\\ \hfill 3& \hfill 1& \hfill -2\\ \hfill 2& \hfill -4& \hfill 6\end{array}\text{ }|\text{ }\begin{array}{rr}\hfill 1& \hfill -2\\ \hfill 3& \hfill 1\\ \hfill 2& \hfill -4\end{array}|[/latex]

Then,

[latex]1\left(1\right)\left(6\right)+\left(-2\right)\left(-2\right)\left(2\right)+3\left(3\right)\left(-4\right)-2\left(1\right)\left(3\right)-\left(-4\right)\left(-2\right)\left(1\right)-6\left(3\right)\left(-2\right)=0[/latex]

As the determinant equals zero, there is either no solution or an infinite number of solutions. We have to perform elimination to find out.

1. Multiply equation (1) by [latex]-2[/latex] and add the result to equation (3):
   [latex]\frac{\begin{array}{r}\hfill -2x+4y - 6x=0\\ \hfill 2x - 4y+6z=0\end{array}}{0=0}[/latex]
2. Obtaining an answer of [latex]0=0[/latex], a statement that is always true, means that the system has an infinite number of solutions. Graphing the system, we can see that two of the planes are the same and they both intersect the third plane on a line.

Figure 6

 Understanding Properties of Determinants

Key Concepts

• The determinant for [latex]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right][/latex] is [latex]ad-bc[/latex].
• Cramer’s Rule replaces a variable column with the constant column. Solutions are [latex]x=\frac{{D}_{x}}{D},y=\frac{{D}_{y}}{D}[/latex].
• To find the determinant of a 3×3 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right).
• To solve a system of three equations in three variables using Cramer’s Rule, replace a variable column with the constant column for each desired solution: [latex]x=\frac{{D}_{x}}{D},y=\frac{{D}_{y}}{D},z=\frac{{D}_{z}}{D}[/latex].
• Cramer’s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions.
• Certain properties of determinants are useful for solving problems. For example:
  • If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.
  • When two rows are interchanged, the determinant changes sign.
  • If either two rows or two columns are identical, the determinant equals zero.
  • If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.
  • The determinant of an inverse matrix [latex]{A}^{-1}[/latex] is the reciprocal of the determinant of the matrix [latex]A[/latex].
  • If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.

Glossary

Cramer’s Rule

a method for solving systems of equations that have the same number of equations as variables using determinants

determinant

a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations

Section Exercises

Licenses & Attributions