Solve Systems of Equations Using Determinants

In this section we will learn of another method to solve systems of linear equations called Cramer’s Rule. Before we can begin to use the rule, we need to learn some new definitions and notation.

Evaluate the determinant of a 2 × 2 matrix

If a matrix has the same number of rows and columns, we call it a square matrix. Each square matrix has a real number associated with it called its determinant. To find the determinant of the square matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, we first write it as $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$. To get the real number value of the determinant we subtract the products of the diagonals, as shown.

Example. Evaluate the determinant of (a) $\begin{vmatrix} 4 & -2 \\ 3 & -1 \end{vmatrix}$ (b) $\begin{vmatrix} -3 & -4 \\ -2 & 0 \end{vmatrix}$.

(a) We subtract the products of the diagonals:

(b)

Evaluate the determinant of a 3 × 3 matrix

To evaluate the determinant of a $3 \times 3$ matrix, we have to be able to evaluate the minor of an entry in the determinant. The minor of an entry is the $2 \times 2$ determinant found by eliminating the row and column in the $3 \times 3$ determinant that contains the entry.

To find the minor of entry $a_1$ in $\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$, we eliminate the row and column that contain it — the first row and first column — and write the $2 \times 2$ determinant that remains:

To find the minor of entry $b_2$, we eliminate the row and column that contain it — the second row and second column — and write the $2 \times 2$ determinant that remains:

Example. For the determinant $\begin{vmatrix} 4 & -2 & 3 \\ 1 & 0 & -3 \\ -2 & -4 & 2 \end{vmatrix}$, find and then evaluate the minor of (a) $a_1$ (b) $b_3$ (c) $c_2$.

(a) We eliminate the row and column that contain $a_1$ (the first row and first column):

(b) We eliminate the row and column that contain $b_3$ (the third row and second column):

(c) We eliminate the row and column that contain $c_2$ (the second row and third column):

We are now ready to evaluate a $3 \times 3$ determinant. To do this we expand by minors, which allows us to evaluate the $3 \times 3$ determinant using $2 \times 2$ determinants — which we already know how to evaluate! To evaluate a $3 \times 3$ determinant by expanding by minors along the first row, we use the following pattern.

Example. Evaluate the determinant $\begin{vmatrix} 2 & -3 & -1 \\ 3 & 2 & 0 \\ -1 & -1 & -2 \end{vmatrix}$ by expanding by minors along the first row.

To evaluate a $3 \times 3$ determinant we can expand by minors using any row or column. Choosing a row or column other than the first row sometimes makes the work easier. When we expand by any row or column, we must be careful about the sign of the terms in the expansion. To determine the sign of the terms, we use the following sign pattern chart.

Since we can expand by any row or column, how do we decide which one to use? Usually we try to pick a row or column that will make our calculation easier. If the determinant contains a $0$, using the row or column that contains the $0$ will make the calculations easier.

Example. Evaluate the determinant $\begin{vmatrix} 4 & -1 & -3 \\ 3 & 0 & 2 \\ 5 & -4 & -3 \end{vmatrix}$ by expanding by minors.

To expand by minors, we look for a row or column that will make our calculations easier. Since $0$ is in the second row and second column, expanding by either of those is a good choice. Since the second row has fewer negatives than the second column, we will expand by the second row — watching the $-\,+\,-$ signs for that row:

Use Cramer’s Rule to solve systems of equations

Cramer’s Rule is a method of solving systems of equations using determinants. It can be derived by solving the general form of the systems of equations by elimination. Here we will demonstrate the rule for both systems of two equations with two variables and for systems of three equations with three variables.

Let’s start with the systems of two equations with two variables.

Example. Solve using Cramer’s Rule: $\left\{\begin{array}{l} 2x+y=-4 \\ 3x-2y=-6 \end{array}\right.$

Evaluate the determinant $D$, using the coefficients of the variables:

Evaluate the determinant $D_x$. Replace the coefficients of $x$, $2$ and $3$, with the constants, $-4$ and $-6$:

Evaluate the determinant $D_y$. Replace the coefficients of $y$, $1$ and $-2$, with the constants, $-4$ and $-6$:

Substitute in the values of $D$, $D_x$, and $D_y$ to find $x$ and $y$:

The solution is the ordered pair $(-2,0)$. Substituting $x=-2,\ y=0$ into both original equations confirms $(-2,0)$ is the solution to the system.

To solve a system of three equations with three variables with Cramer’s Rule, we basically do what we did for a system of two equations. However, we now have to solve for three variables to get the solution. The determinants are also going to be $3 \times 3$ which will make our work more interesting!

Example. Solve the system of equations using Cramer’s Rule: $\left\{\begin{array}{l} 3x-5y+4z=5 \\ 5x+2y+z=0 \\ 2x+3y-2z=3 \end{array}\right.$

Evaluate the determinant $D$, using the coefficients of the variables, by expanding by minors using column $1$:

Evaluate the determinant $D_x$. Use the constants to replace the coefficients of $x$, then expand by minors using column $1$:

Evaluate the determinant $D_y$. Use the constants to replace the coefficients of $y$, then expand by minors using column $2$:

Evaluate the determinant $D_z$. Use the constants to replace the coefficients of $z$, then expand by minors using column $3$:

Find $x$, $y$, and $z$, then write the solution as an ordered triple:

The solution is $(2,-3,-4)$. We leave the check, that this ordered triple is a solution to all three original equations, to you.

Cramer’s Rule does not work when the value of the determinant $D$ is $0$, as this would mean we would be dividing by $0$. But when $D=0$, the system is either inconsistent or dependent.

When the value of $D=0$ and $D_x$, $D_y$, and $D_z$ are all zero, the system is consistent and dependent and there are infinitely many solutions.

When the value of $D=0$ and $D_x$, $D_y$, and $D_z$ are not all zero, the system is inconsistent and there is no solution.

In the next example, we will use the values of the determinants to find the solution of the system.

Example. Solve the system of equations using Cramer’s Rule: $\left\{\begin{array}{l} x+3y=4 \\ -2x-6y=3 \end{array}\right.$

Evaluate the determinant $D$, using the coefficients of the variables:

We cannot use Cramer’s Rule to solve this system. But by looking at the value of the determinants $D_x$ and $D_y$, we can determine whether the system is dependent or inconsistent.

Since all the determinants are not zero, the system is inconsistent. There is no solution.

Solve applications using determinants

An interesting application of determinants allows us to test if points are collinear. Three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$ are collinear if and only if the determinant below is zero.

We will use this property in the next example.

Example. Determine whether the points $(5,-5)$, $(4,-3)$, and $(3,-1)$ are collinear.

Substitute the values into the determinant, then evaluate it by expanding by minors using column $3$:

The value of the determinant is $0$, so the points are collinear.

Key terms

square matrix — a matrix that has the same number of rows and columns. determinant — a real number associated with a square matrix; for $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, its value is $\begin{vmatrix} a & b \\ c & d \end{vmatrix}=ad-bc$. minor of an entry — in a $3\times 3$ determinant, the $2\times 2$ determinant found by eliminating the row and column that contain the entry. Cramer’s Rule — a method of solving a system of equations using the determinant of the coefficient matrix, $D$, and the determinants $D_x$, $D_y$ (and $D_z$) formed by replacing one column of coefficients with the constants; $x=\tfrac{D_x}{D}$, and so on. collinear points — points that lie on the same line; three points are collinear if and only if the $3\times 3$ determinant formed from their coordinates and a column of $1\text{s}$ is zero.

This section is adapted from Intermediate Algebra 2e, Section 4.6: Solve Systems of Equations Using Determinants by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: represented determinants and the sign-pattern chart with KaTeX vmatrix notation instead of the source’s crossed-out row/column diagrams, and its step tables as single display equality chains; recreated the Dependent and Inconsistent Systems table as a markdown table; omitted the Be Prepared quiz, Media links, self-check, and section exercises; and converted the source Try Its into interactive exercises with instant feedback, reducing multi-part items to a single part each.