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 $2\times 2$

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 $\left[\begin{array}{c}ab\hfill \\ cd\hfill \end{array}\right],$

we first write it as $\backslash |\begin{array}{c}ab\hfill \\ cd\hfill \end{array}\backslash |.$

To get the real number value of the determinate we subtract the products of the diagonals, as shown.

<div data-type="note" markdown="1">

The determinant of any square matrix $\left[\begin{array}{c}ab\hfill \\ cd\hfill \end{array}\right],$

where a, b, c, and d are real numbers, is

</div>

Evaluate the Determinant of a $3\times 3$

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 $a_1,$

we eliminate the row and column which contain it. So we eliminate the first row and first column. Then we 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. So we eliminate the 2^nd row and 2^nd column. Then we write the $2\times 2$

determinant that remains.

<div data-type="example">

For the determinant $\backslash |\begin{array}{ccccccc}\hfill 4& & & \hfill \mathrm{-2}& & & \hfill 3\\ \hfill 1& & & \hfill 0& & & \hfill \mathrm{-3}\\ \hfill \mathrm{-2}& & & \hfill \mathrm{-4}& & & \hfill 2\end{array}\backslash |,$

find and then evaluate the minor of ⓐ $a_1$

ⓑ $b_3$

ⓒ $c_2.$

ⓐ* * *

{: valign=”top”}	Eliminate the row and column that contains $a_1.$

{: valign=”top”}	Write the $2\times 2$

determinant that remains. | | {: valign=”top”}| Evaluate. | | {: valign=”top”}| Simplify. | | {: valign=”top”}{: .unnumbered .unstyled summary=”The first row of the 3 by 3 determinant is 4, minus 2, 3. Row 2 is 1, 0, minus 3. Row 3 is minus 2, minus 4, 2. Eliminating the row and column containing a1, we get the minor of a1. This 2 by 2 determinant has row 1: 0, minus 3 and row 2: minus 4, 2. Evaluate and simplify to get minus 12.” data-label=””}

---

---

ⓑ* * *

Eliminate the row and column that contains $b_3.$

{: valign=”top”}	Write the $2\times 2$

determinant that remains. | | {: valign=”top”}| Evaluate. | | {: valign=”top”}| Simplify. | | {: valign=”top”}{: .unnumbered .unstyled summary=”The 3 by 3 matrix has row 1: 4, minus 2, 3, row 2: 1, 0, minus 3 and row 3: minus 2, minus 4, 2. Eliminating the row and column containing b3, we get minor of b3 with row 1: 4, 3 and row 2: 1, minus 3. Evaluate and simplify to get minus 15.” data-label=””}

---

---

ⓒ* * *

{: valign=”top”}	Eliminate the row and column that contains $c_2.$

{: valign=”top”}	Write the $2\times 2$

determinant that remains. | | {: valign=”top”}| Evaluate. | | {: valign=”top”}| Simplify. | | {: valign=”top”}{: .unnumbered .unstyled .can-break summary=”The 3 by 3 matrix has row 1: 4, minus 2, 3, row 2: 1, 0, minus 3 and row 3: minus 2, minus 4, 2. Eliminating the row and column containing c2, we get minor of c2 with row 1: 4, minus 2 and row 2: minus 2, 4. Evaluate and simplify to get 12” data-label=””}

</div>

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:

Remember, to find the minor of an entry we eliminate the row and column that contains the entry.

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.

Notice that the sign pattern in the first row matches the signs between the terms in the expansion by the first row.

Since we can expand by any row or column, how do we decide which row or column 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.

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.

Notice that to form the determinant D, we use take the coefficients of the variables.

Notice that to form the determinant $D_x$

and $D_y,$

we substitute the constants for the coefficients of the variable we are finding.

<div data-type="example">

</div>

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!

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

Evaluate the determinant D.
Expand by minors using column 1.
Evaluate the determinants.
Simplify.
Simplify.
Simplify.
Evaluate the determinant $D_x.$ Use the constants to replace the coefficients of x.
Expand by minors using column 1.
Evaluate the determinants.
Simplify.
Simplify.
Evaluate the determinant $D_y.$ Use the constants to replace the coefficients of y.
Evaluate the determinants.
Simplify.
Simplify.
Simplify.
Evaluate the determinant $D_z.$ Use the constants to replace the coefficients of z.
Evaluate the determinants.
Simplify.
Simplify.
Simplify.
Find x, y, and z.
Substitute in the values.
Simplify.
Write the solution as an ordered triple.
Check that the ordered triple is a solution to all three original equations.	We leave the check to you.
	The solution is $\left(2,\mathrm{-3},\mathrm{-4}\right).$

Cramer’s rule does not work when the value of the D determinant 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.

Solve Applications using Determinants

An interesting application of determinants allows us to test if points are collinear. Three points $\left(x_1,y_1\right),$

and $\left(x_3,y_3\right)$

are collinear if and only if the determinant below is zero.

We will use this property in the next example.

Determine whether the points $\left(5,\mathrm{-5}\right),$

$\left(4,\mathrm{-3}\right),$

and $\left(3,\mathrm{-1}\right)$

are collinear.

Substitute the values into the determinant. $\left(5,\mathrm{-5}\right),$ $\left(4,\mathrm{-3}\right),$ and $\left(3,\mathrm{-1}\right)$
Evaluate the determinant by expanding by minors using column 3.
Evaluate the determinants.
Simplify.
Simplify.
	The value of the determinate is 0, so the points are collinear.

Key Concepts

• Determinant: The determinant of any square matrix $\left[\begin{array}{c}ab\hfill \\ cd\hfill \end{array}\right],$

  where a, b, c, and d are real numbers, is

  ---

  $\backslash |\begin{array}{c}ab\hfill \\ cd\hfill \end{array}\backslash |=ad-bc$
• Expanding by Minors along the First Row to Evaluate a 3 × 3 Determinant: To evaluate a $3\times 3$

  determinant by expanding by minors along the first row, the following pattern:

  ---

  

• Sign Pattern: When expanding by minors using a row or column, the sign of the terms in the expansion follow the following pattern.

  ---

  $\backslash |\begin{array}{c}+-+\hfill \\ -+-\hfill \\ +-+\hfill \end{array}\backslash |$
• Cramer’s Rule: For the system of equations $\{\begin{array}{c}{a}_{1}x+b_{1}y=k_1\hfill \\ a_{2}x+b_{2}y=k_2\hfill \end{array},$

  the solution

  $\left(x,y\right)$

  can be determined by

  ---

  

  ---

  Notice that to form the determinant D, we use take the coefficients of the variables.

• How to solve a system of two equations using Cramer’s rule.
  1. Evaluate the determinant D, using the coefficients of the variables.
  2. Evaluate the determinant $D_x.$

     Use the constants in place of the x coefficients.

  3. Evaluate the determinant $D_y.$

     Use the constants in place of the y coefficients.

  4. Find x and y. $x=\frac{D_x}{D},$ $y=\frac{D_y}{D}.$
  5. Write the solution as an ordered pair.
  6. Check that the ordered pair is a solution to both original equations.
  7. Dependent and Inconsistent Systems of Equations: For any system of equations, where the value of the determinant $D=0,$

     ---

     $\begin{array}{ccccccc}\mathbf{\text{Value of determinants}}\hfill & & & \mathbf{\text{Type of system}}\hfill & & & \mathbf{\text{Solution}}\hfill \\ D=0\text{and}{D}_x,D_y\text{and}{D}_z\text{are all zero}\hfill & & & \text{consistent and dependent}\hfill & & & \text{infinitely many solutions}\hfill \\ D=0\text{and}{D}_x,D_y\text{and}{D}_z\text{are not all zero}\hfill & & & \text{inconsistent}\hfill & & & \text{no solution}\hfill \end{array}$
  8. Test for Collinear Points: Three points $\left(x_1,y_1\right),$ $\left(x_2,y_2\right),$

     and

     $\left(x_3,y_3\right)$

     are collinear if and only if

     ---

     $\backslash |\begin{array}{c}{x}_1{y}_{1}1\hfill \\ x_2{y}_{2}1\hfill \\ x_3{y}_{3}1\hfill \end{array}\backslash |=0$

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