Algebra and Trigonometry, 5th Edition

9.5 Inverses of Matrices

• Find the inverse of a square matrix, if it exists.

• Use inverses of matrices to solve systems of equations.

In this section, we continue our study of matrix algebra, finding the multiplicative inverse, or simply inverse, of a square matrix, if it exists. Then we use such inverses to solve systems of equations.

The Identity Matrix

JUST IN TIME 2

Recall that, for real numbers, $a·1=1·a=a$; 1 is the multiplicative identity. A multiplicative identity matrix is very similar to the number 1.

Identity Matrix

For any positive integer n, the $n\times n$ identity matrix is an $n\times n$ matrix with 1’s on the main diagonal and 0’s elsewhere and is denoted by

$$\mathbf{\text{I}}=\left[\begin{array}{ccc}1& 0& 0\dots 0\\ 0& 1& 0\dots 0\\ 0& 0& 1\dots 0\\ ⋮& ⋮& ⋮⋮\\ 0& 0& 0\dots 1\end{array}\right]$$

Then $\mathbf{AI}=\mathbf{IA}=\mathbf{\text{A}}$, for any $n\times n$ matrix A.

Example