Find the Inverse of a 3 by 3 Matrix

Calculate the inverse of a 3x3 matrix using minors, cofactors, adjugate, and determinant. Generate random matrices and see detailed step-by-step solutions.

📐 3x3 Matrix Inverse Calculator

Step-by-step: determinant → minors → cofactors → adjugate → inverse

✨ To find the inverse of a 3x3 matrix \( A \), we compute the determinant \( D \). If \( D \neq 0 \), the inverse exists and is given by \( A^{-1} = \frac{1}{D} \text{adj}(A) \), where \( \text{adj}(A) \) is the adjugate (transpose of the cofactor matrix).
\( A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 0 & 2 \\ -2 & 4 & 1 \end{bmatrix} \)
Find the inverse matrix. Click "Show me" for each step.

📖 Step-by-Step Solution

STEP 1: Calculate the determinant D of the given matrix
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STEP 2: Determine the matrix of minors M
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STEP 3: Determine the matrix of cofactors C (using sign chart)
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STEP 4: Determine the adjugate matrix (transpose of cofactor matrix)
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STEP 5: Calculate the inverse using \( A^{-1} = \frac{1}{D} \text{adj}(A) \)
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