Inverse Matrix Calculator

Inverse Matrix Calculator

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.

Choose the size of the matrix:

Method:

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. Let's explore how it works!

What is the Inverse of a Matrix?

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 matrix must be square.
The determinant |A| ≠ 0.

The inverse matrix calculator can quickly verify these conditions and compute the inverse efficiently. For a 3x3 matrix, let $$ A = \begin{bmatrix}a & b & c\\ d & e & f\\ g & h & i\end{bmatrix} $$

Then 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:

The minor of an element is the determinant obtained by removing the row and column containing that element.

Cofactor:

The cofactor of an element is the minor multiplied by (-1)^(i+j), where i and j are the row and column indices: $$ Cofactor(a_{ij}) = (-1)^{i+j} \times Minor(a_{ij}) $$

For the 3x3 matrix above, the cofactors are:

$$ M_{11} = (-1)^{1+1} \begin{vmatrix} e & f\\ h & i \end{vmatrix}, \quad M_{12} = (-1)^{1+2} \begin{vmatrix} d & f\\ g & i \end{vmatrix}, \quad ... $$

The cofactor matrix is then transposed to obtain the adjugate. Using the inverse matrix calculator simplifies all these calculations.

Determinant:

The determinant is a single number representing the matrix. It can be calculated as the sum of the products of elements and their cofactors along any row or column. The determinant calculator can quickly compute this.

Singular Matrix:

A matrix with determinant zero (|A| = 0) is singular and cannot be inverted. The inverse matrix calculator identifies singular matrices automatically.

Non-Singular Matrix:

A matrix with |A| ≠ 0 is non-singular or invertible, meaning its inverse can be calculated.

Gauss-Jordan Method:

The Gauss-Jordan method transforms the 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 performing row operations to convert the left side into the identity matrix, the right side becomes the inverse of the original matrix. The calculator performs this instantly.

Example:

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: $$ \begin{bmatrix}3 & 1 & -4\\ -1.182 & -0.182 & 1.545\\ -0.091 & -0.091 & 0.273\end{bmatrix} $$

Working of the Inverse Matrix Calculator:

The calculator simplifies inverse matrix computation, making it quick and easy to get results.

Input:

Select the matrix size from the dropdown.
Enter matrix values and click “Generate Matrix.”
Choose the method to compute the inverse.
Click “Calculate.”

Output:

The inverse matrix is displayed along with the original matrix values.
All calculation steps are shown for learning purposes.

FAQs:

What is an Invertible Matrix?

A square matrix with a non-zero determinant is invertible (non-singular) and has an inverse.

Can we find the inverse of all matrices?

No. Only non-singular (invertible) matrices have an inverse. The calculator checks this automatically.

Can I obtain the original matrix after inversion?

Yes. Enter the inverted matrix and calculate its inverse to recover the original matrix.

Can you invert a singular matrix?

No. Singular matrices (determinant = 0) do not have an inverse. The calculator can identify singular matrices instantly.

Conclusion:

Finding the inverse of a matrix is essential for solving linear systems. While manual computation for 3x3 or 4x4 matrices is lengthy, the inverse matrix calculator performs it efficiently with all steps shown.

References:

Wikipedia: Invertible matrix, properties.
Embibe: Inverse Matrix, meaning, formula, examples.