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Write down entries of the matrix and the calculator will find its inverse by applying various methods to it, with step-by-step calculations shown.
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The free online inverse matrix calculator allows you to compute the inverse of 2x2, 3x3, or higher-order square matrices quickly. You can also learn how to find the inverse using the Gauss-Jordan method or the Adjugate method with step-by-step guidance.
The inverse of a matrix is mathematically defined as:
$$ A^{-1} = \frac{Adj(A)}{|A|} $$
Where:
For a 2x2 matrix:
$$ A = \begin{bmatrix}a & b\\ c & d \end{bmatrix}, \quad Adj(A) = \begin{bmatrix}d & -b\\ -c & a \end{bmatrix}, \quad |A| = ad - bc $$
Conditions for a matrix to have an inverse:
The calculator verifies these conditions and computes the inverse efficiently.
Let $$ A = \begin{bmatrix}a & b & c\\ d & e & f\\ g & h & i\end{bmatrix} $$
The adjugate is:
$$ Adj(A) = \begin{bmatrix}M_{11} & M_{12} & M_{13}\\ M_{21} & M_{22} & M_{23}\\ M_{31} & M_{32} & M_{33}\end{bmatrix}^{T} $$
where minors and cofactors are used to determine each element.
Minor: Determinant obtained by removing the row and column of the element.
Cofactor: $$ Cofactor(a_{ij}) = (-1)^{i+j} \times Minor(a_{ij}) $$
Example for the 3x3 matrix:
$$ M_{11} = (-1)^{1+1} \begin{vmatrix} e & f\\ h & i \end{vmatrix}, \quad M_{12} = (-1)^{1+2} \begin{vmatrix} d & f\\ g & i \end{vmatrix}, \dots $$
The determinant represents a single value for the matrix. It can be calculated as the sum of products of elements and their cofactors along any row or column. The determinant calculator computes this instantly.
The Gauss-Jordan method transforms a matrix into the identity matrix using row operations:
$$ \left[\begin{array}{ccc|ccc} a & b & c & 1 & 0 & 0\\ d & e & f & 0 & 1 & 0\\ g & h & i & 0 & 0 & 1 \end{array}\right] $$
After row operations, the right-hand side becomes the inverse matrix.
Find the inverse of:
$$ \begin{bmatrix}1 & 1 & 9\\ 2 & 5 & 1\\ 1 & 2 & 7 \end{bmatrix} $$
Using Gauss-Jordan elimination:
$$ \left[\begin{array}{ccc|ccc} 1 & 1 & 9 & 1 & 0 & 0\\ 2 & 5 & 1 & 0 & 1 & 0\\ 1 & 2 & 7 & 0 & 0 & 1 \end{array}\right] $$
The final inverse matrix is:
$$ \begin{bmatrix}3 & 1 & -4\\ -1.182 & -0.182 & 1.545\\ -0.091 & -0.091 & 0.273\end{bmatrix} $$
Input:
Output:
A square matrix with a non-zero determinant is invertible (non-singular).
No. Only non-singular matrices have an inverse. The calculator checks this automatically.
Yes. Enter the inverted matrix and compute its inverse to recover the original.
No. Singular matrices (determinant = 0) do not have an inverse. The calculator identifies them instantly.
Finding the inverse of a matrix is essential for solving linear systems. While manual computation for 3x3 or 4x4 matrices is time-consuming, the inverse matrix calculator performs it efficiently with all steps displayed.
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