Problem 32 - Precalculus with Limits

Question

In Exercises \(25-34\) , use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$\left[\begin{array}{rrr}{0.6} & {0} & {-0.3} \\ {0.7} & {-1} & {0.2} \\\ {1} & {0} & {-0.9}\end{array}\right]$$

Step-by-Step Solution

Answer

To find the inverse of a matrix, first make sure the matrix is invertible. Then, calculate the minors, make the matrix of minors, calculate the cofactors, create the matrix of cofactors, transpose it to get the adjugate or adjoint of the matrix, and finally, divide the adjugate by the determinant of the original matrix.

Step 1: Verify if the matrix is invertible

For a matrix to have an inverse, its determinant must not equal 0. First, calculate the determinant of the given matrix. If the determinant equals 0, the matrix is not invertible; if it doesn't, continue with the following steps.

Step 2: Calculate the minor of each element in the matrix

Identify all 2x2 submatrices within the 3x3 matrix by eliminating one row and one column that correspond to each element. Then find the determinant of each submatrix. For instance, the minor of the element at the first row and first column (0.6) is calculated by deleting the first row and first column and finding the determinant of the resulting 2x2 matrix.

Step 3: Build the matrix of minors

Replace each element in the main matrix with the corresponding minor obtained in Step 2. This will generate the matrix of minors.

Step 4: Create the matrix of cofactors

For a 3x3 matrix, each cofactor equals the minor with its sign adjusted according to the rule: Add the row number and the column number of the element. If the result is even, the cofactor is the same as the minor; if it's odd, the cofactor is the negative of the minor. This rule is known as checkerboard of pluses and minuses.

Step 5: Find the transpose of the matrix of cofactors

Transpose the matrix of cofactors. This is achieved by interchanging its rows and columns. The resulting matrix is known as the adjugate or adjoint of the original matrix.

Step 6: Calculate the inverse of the matrix

The inverse of the matrix is the adjugate matrix divided by the determinant of the original matrix. Each element of the adjugate matrix is divided by the determinant to form the inverse matrix.