To find the inverse of a 3x3 matrix, you can use the following steps, provided the matrix is invertible (i.e., its determinant is non-zero):

1. **Find the Determinant**: Calculate the determinant \( \text{det}(A) \) of the 3x3 matrix \( A \). For a matrix 
   \[
   A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}
   \]
   the determinant is calculated as:
   \[
   \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
   \]

2. **Calculate the Matrix of Minors**: For each element in the matrix, calculate the determinant of the 2x2 submatrix that remains after removing the row and column of that element. This will give you a new 3x3 matrix, known as the matrix of minors.

3. **Form the Cofactor Matrix**: Apply a checkerboard pattern of signs (+, -, + in the first row, etc.) to the matrix of minors to create the cofactor matrix.

4. **Transpose the Cofactor Matrix**: This gives the adjugate (or adjoint) matrix.

5. **Divide by the Determinant**: Finally, multiply each element in the adjugate matrix by \( \frac{1}{\text{

Learn the step-by-step process to calculate the inverse of a 3x3 matrix. This guide covers key concepts such as determinants, matrix of minors, cofactor matrix, adjugate matrix, and the formula for matrix inversion. Suitable for high school students in Grades 11-12.

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