Inverse Matrix 3×3 Calculator
This calculator finds the inverse of a 3×3 matrix using the adjoint method or row operations. It is essential for students, engineers, and professionals solving matrix equations, linear systems, or transformations in linear algebra.
3×3 Matrix Inverse Solver
Inverse of 3×3 Matrix Formula
Where:
- $$A$$ is the 3×3 matrix
- $$\det(A)$$ is the determinant of $$A$$
- $$\operatorname{adj}(A)$$ is the adjugate (transpose of cofactor matrix) of $$A$$
The matrix $$A$$ is invertible only if $$\det(A) \neq 0$$.
Inverse Matrix 3×3 – Calculation Example
$$A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}$$
$$\det(A) = 1(1 \cdot 0 – 4 \cdot 6) – 2(0 \cdot 0 – 4 \cdot 5) + 3(0 \cdot 6 – 1 \cdot 5) = 1(0 – 24) – 2(0 – 20) + 3(0 – 5) = -24 + 40 – 15 = 1$$
Since \(\det(A) = 1\), the inverse exists and:
$$A^{-1} = \operatorname{adj}(A)$$
The inverse of a 3×3 matrix is used in solving systems of linear equations, finding transformations, and applications in physics and engineering. Our calculator automates determinant, cofactor, and adjugate computations for accuracy and speed.