Find the inverse, if it exists, for each matrix. [2x2 matrix]

Verified step by step guidance

1. Step 1: Given a 2x2 matrix A = \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), check if the inverse exists by calculating the determinant of the matrix.
2. Step 2: Calculate the determinant of matrix A using the formula: \( \text{det}(A) = ad - bc \).
3. Step 3: If the determinant is not equal to zero (\( \text{det}(A) \neq 0 \)), the inverse exists. Otherwise, the matrix does not have an inverse.
4. Step 4: If the inverse exists, use the formula for the inverse of a 2x2 matrix: \( A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).
5. Step 5: Substitute the values of a, b, c, and d from the given matrix into the inverse formula to find the inverse matrix.

Key Concepts

Matrix Inversion

Matrix inversion is the process of finding a matrix that, when multiplied with the original matrix, yields the identity matrix. For a 2x2 matrix, the inverse can be calculated using the formula A^(-1) = (1/det(A)) * adj(A), where det(A) is the determinant and adj(A) is the adjugate of the matrix. An inverse exists only if the determinant is non-zero.

Determinant

The determinant is a scalar value that provides important information about a matrix, including whether it is invertible. For a 2x2 matrix represented as [[a, b], [c, d]], the determinant is calculated as det(A) = ad - bc. If the determinant equals zero, the matrix does not have an inverse, indicating that the rows (or columns) are linearly dependent.

Adjugate of a Matrix

The adjugate (or adjoint) of a matrix is the transpose of its cofactor matrix. For a 2x2 matrix, the adjugate can be found by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. The adjugate is used in the formula for finding the inverse of a matrix, particularly when the determinant is non-zero.