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Chapter 6: Problem 50

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \( \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).
03

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.

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Most popular questions from this chapter

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