Problem 59 - College Algebra

Question

Explain how to find the multiplicative inverse for a \(3 \times 3\) invertible matrix.

Step-by-Step Solution

Answer

The inverse of a \(3 \times 3\) matrix can be found by setting up an augmented matrix with the given matrix and an identity matrix, performing row operations to turn the given matrix into an identity matrix. Once this is achieved, the remaining right side of the augmented matrix is the inverse of the original matrix.

Step 1: Set Up the Augmented Matrix

Given an invertible \(3 \times 3\) matrix \(A\), the first step is to set up an augmented matrix. This is a \(3 \times 6\) matrix. The first three columns are the elements of the matrix \(A\) and the last three columns are an identity matrix \(I\) (which has 1's down the main diagonal and 0's elsewhere).

Step 2: Perform Row Operations

The next step involves the performance of a series of row operations to turn the left half of the augmented matrix (which is Matrix \(A\)) into the identity matrix. The operations that you can use include swapping rows, multiplying a row by a constant, or replacing one row with the sum of the row and a multiple of another row.

Step 3: Obtain the Inverse Matrix

Once the identity matrix is obtained on the left side of the augmented matrix, the right side of the augmented matrix is the inverse of the original matrix \(A\). The inverse is denoted as \(A^{-1}\)