Problem 50 - College Algebra

Question

Explain how to find the multiplicative inverse for a \(2 \times 2\) invertible matrix.

Step-by-Step Solution

Answer

The inverse of a \(2 \times 2\) matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is given by \(A^{-1} = \frac{1}{ad-bc} \cdot \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \), if \(ad - bc \neq 0\).

Step 1: Compute the Determinant of the Matrix

The first step in finding the inverse is to compute the determinant of the matrix. The determinant of a \(2 \times 2\) matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is given by \( \text{det}(A) = ad - bc \). It's important to remember that if \( \text{det}(A) = 0 \), then the matrix A does not have an inverse.

Step 2: Find the Adjugate of the Matrix

The adjugate of a \(2 \times 2\) matrix is obtained by swapping the elements \(a\) and \(d\), and changing the sign of elements \(b\) and \(c\). So, the adjugate matrix of matrix A is \( \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).

Step 3: Find the Inverse of the Matrix

Finally, the inverse of the matrix A can be found with the formula \(A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)\). This involves multiplying each element of the adjugate matrix by the reciprocal of the determinant.