Algebra and Trigonometry, 5th Edition

9.6 Determinants and Cramer’s Rule

• Evaluate determinants of square matrices.

• Use Cramer’s rule to solve systems of equations.

Determinants of Square Matrices

With every square matrix, we associate a number called its determinant.

Determinant of a $\mathrm{2 }\times \mathrm{ 2}$ Matrix

The determinant of the matrix $[\begin{array}{cc}a& c\\ b& d\end{array}]$ is denoted $|\begin{array}{cc}a& c\\ b& d\end{array}|$ and is defined as $\left|\begin{array}{cc}a& c\\ b& d\end{array}\right|=ad-bc$.

JUST IN TIME 22

Example 1

Evaluate: $\left|\begin{array}{cc}\hfill \sqrt{2}& \hfill -3\\ \hfill -4& \hfill -\sqrt{2}\end{array}\right|$.

Solution

$\begin{array}{c}=\sqrt{2}(-\sqrt{2})-(-4)(-3)\hfill \\ =-2-12\hfill \\ =-14\hfill \end{array}$

Now Try Exercise 1.

We now consider a way to evaluate determinants of square matrices of order $\mathrm{3 }\times \mathrm{ 3}$ or higher.

Evaluating Determinants Using Cofactors

Often we first find minors and cofactors of matrices in ...