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 \( \det(A) \) is nonzero. If it is zero, the inverse does not exist.
4. If the determinant is nonzero, find the inverse matrix using the formula \( A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).
5. Write the inverse matrix explicitly by substituting the values of \( a, b, c, d \) and the determinant into the formula.

Key Concepts

Matrix Inverse

The inverse of a matrix A is another matrix, denoted A⁻¹, such that when multiplied together, they yield 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. It indicates whether the matrix is invertible; if the determinant is zero, the matrix has no inverse. The determinant also provides information about the matrix's scaling and orientation effects.

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.