Find the inverse, if it exists, for each matrix.

Verified step by step guidance

1. Identify the given 2x2 matrix as \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
2. Calculate the determinant of matrix \( A \) using the formula \( \det(A) = ad - bc \).
3. Check if the determinant is nonzero. If \( \det(A) \neq 0 \), the inverse exists; otherwise, the inverse does not exist.
4. If the inverse exists, use the formula for the inverse of a 2x2 matrix: \( A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).
5. Substitute the values of \( a, b, c, d \) and the determinant into the formula to express the inverse matrix.

Key Concepts

Matrix Inverse

The inverse of a matrix A is another matrix, denoted A⁻¹, such that when multiplied together, they produce the identity matrix. Only square matrices with nonzero determinants have inverses. Finding the inverse is essential for solving matrix equations and understanding linear transformations.

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This scalar value indicates whether the matrix is invertible; if the determinant is zero, the matrix has no inverse. The determinant also provides insight into the matrix's scaling effect on area.

Formula for the Inverse of a 2x2 Matrix

For a 2x2 matrix [[a, b], [c, d]] with nonzero determinant, the inverse is (1/det) times the matrix [[d, -b], [-c, a]]. This formula swaps the diagonal elements, changes the signs of the off-diagonal elements, and scales by the reciprocal of the determinant, providing a straightforward method to find the inverse.