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

Verified step by step guidance

1. Identify the given 3x3 matrix \( A \) for which you need to find the inverse.
2. Recall that the inverse of a matrix \( A \), denoted \( A^{-1} \), exists only if \( \det(A) \neq 0 \). So, calculate the determinant of \( A \) using the formula for a 3x3 matrix determinant:
3. \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] where the matrix is \( \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} $$.
4. If \( \det(A) \neq 0 \), proceed to find the matrix of minors, then convert it to the matrix of cofactors by applying the checkerboard pattern of signs.
5. Transpose the matrix of cofactors to get the adjugate matrix \( \text{adj}(A) \), and finally compute the inverse by dividing the adjugate by the determinant:
6. \[ A^{-1} = \frac{1}{\det(A)} \times \text{adj}(A) \].

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 Matrix

The determinant is a scalar value computed from a square matrix that indicates whether the matrix is invertible. A zero determinant means the matrix is singular and has no inverse. For a 3x3 matrix, the determinant is calculated using a specific formula involving minors and cofactors.

Methods for Finding the Inverse

Common methods to find a matrix inverse include the adjoint method, which uses cofactors and the determinant, and row reduction to the identity matrix using elementary row operations. Understanding these methods helps in systematically computing the inverse or determining its non-existence.