Find the inverse, if it exists, for each matrix. [3x3 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, first calculate the determinant of \( A \) using the formula for a 3x3 matrix:
3. \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] where the matrix \( A = \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. \[ \begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix} \]
6. Next, transpose the matrix of cofactors to get the adjugate matrix \( \text{adj}(A) \), and finally compute the inverse by dividing the adjugate matrix by the determinant:
7. \[ A^{-1} = \frac{1}{\det(A)} \times \text{adj}(A) \]

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 non-zero 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. If the determinant is zero, the matrix does not have an inverse. Calculating the determinant is a crucial step before attempting to find the inverse.

Methods for Finding the Inverse

Common methods to find a matrix inverse include using the adjoint formula, row reduction to the identity matrix, or applying elementary row operations. For a 3x3 matrix, row reduction or the adjoint method are practical approaches to compute the inverse if it exists.