## What is a determinant?

A determinant is a function whose input is a square matrix and whose output is a number. If $$A$$ is a square matrix, then the determinant of $$A$$ is represented by either $$| A |$$ or $$\det ( A )$$.

Do not confound the determinant symbol with absolute value. A determinant can be a negative number.

For a $$2 \times 2$$ matrix, the determinant is $$\begin{vmatrix} a_1 & b_1\\ a_2 & b_2 \end{vmatrix}=a_1b_2-a_2b_1.$$

For a $$2\times 2$$ matrix, the determinant equals the product of the two numbers on the main diagonal minus the product of the two numbers on the minor diagonal.

## Exercise 7.5.1

Compute the determinant:

$$\begin{vmatrix} 3 & ~~~5\\ 1 & -2 \end{vmatrix}$$

See Solution

## Computing the determinant of a $$3\times3$$ matrix

Let $$A=\begin{vmatrix} a_1 & b_1 & c_1\\a_2 & b_2 & c_2\\a_3 & b_3 & c_3 \end{vmatrix}$$ then

$\det(A)= a_1b_2c_3 +a_2b_3c_1+a_3b_1c_2-a_3b_2c_1-a_2b_1c_3-a_1b_3c_2$

## Exercise 7.5.2

Find the determinant of $$\begin{vmatrix} -1 & 2 & 0\\ \enspace\,3 & 1 & 5\\ -2 & 0 & 1 \end{vmatrix}$$

See Solution

## Cramer's Rule

Cramer's Rule utilizes the coefficient matrix of a system of linear equations and determinants to find the solution.

Consider the general system of two linear equations in two variables:

$$\begin{eqnarray*} a_1x + b_1y &=& c_1\\ a_2x + b_2y &=& c_2 \end{eqnarray*}$$

The coefficient matrix is

$$\left[\begin{array}{rr|r} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ \end{array}\right]$$

Three determinants are defined in terms of the coefficient matrix.

The matrix $$D$$ is formed from the first two columns: $$D=\begin{vmatrix}a_1 & b_1\\a_2 & b_2\end{vmatrix}$$.

The matrix $$D_x$$ is formed from the $$D$$ matrix by replacing the $$x$$-column with the $$c$$-column: $$D_x=\begin{vmatrix}c_1 & b_1\\c_2 & b_2\end{vmatrix}$$.

The matrix $$D_y$$ is formed from the $$D$$ matrix by replacing the $$y$$-column with the $$c$$-column: $$D_y=\begin{vmatrix}a_1 & c_1\\a_2 & c_2\end{vmatrix}$$.

Then Cramer's Rule finds the solution of the system of equations as

$x=\frac{D_x}{D}\qquad y=\frac{D_y}{D}$

## Exercise 7.5.3

Use Cramer's Rule to solve the system:

$$\begin{eqnarray*} 5x + 3y &=& -2\\-3x + ~~y &=& ~~4 \end{eqnarray*}$$

See Solution

## Cramer's Rule applied to a triple system

Consider the general system of three linear equations is three variables:

$$\begin{eqnarray*} a_1x + b_1y +c_1z&=& d_1\\ a_2x + b_2y +c_2z&=& d_2\\ a_3x + b_3y +c_3z&=& d_3\\ \end{eqnarray*}$$

The coefficient matrix is:

$$\left[\begin{array}{rrr|r} a_1 & b_1 & c_1 & d_1\\ a_2 & b_2 & c_2 & d_2\\ a_3 & b_3 & c_3 & d_3 \end{array}\right]$$

The determinants are:

$D= \begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}\quad D_x= \begin{vmatrix} d_1 & b_1 & c_1\\ d_2 & b_2 & c_2\\ d_3 & b_3 & c_3 \end{vmatrix}\quad D_y= \begin{vmatrix} a_1 & d_1 & c_1\\ a_2 & d_2 & c_2\\ a_3 & d_3 & c_3 \end{vmatrix}\quad D_z= \begin{vmatrix} a_1 & b_1 & d_1\\ a_2 & b_2 & d_2\\ a_3 & b_3 & d_3 \end{vmatrix}$

The solution by Cramer's Rule is:

$x=\frac{D_x}{D}\qquad y=\frac{D_y}{D}\qquad z=\frac{D_z}{D}$

## Exercise 7.5.4

Use Cramer's Rule to solve the system:

$$\begin{eqnarray*} 2x -5y + 2z&=& ~~0\\ ~~x + 3y +~~z&=& -1\\ y - 3z &=& ~~0\\ \end{eqnarray*}$$

See Solution