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7.5 Determinants and Cramer's Rule

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