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

Verified step by step guidance

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

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.