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Inverse Matrix Calculator

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The free online inverse matrix calculator computes the inverse matrix of 2×2, 3×3 or higher-order square matrix.
You can learn how to find the inverse of the matrix by the Gauss Jordan and by Adjogate method when using the online calculator.
So let’s move on!

What is the Inverse of a Matrix?

The inverse of a matrix is given as under:

$$ A^{-1} = \frac{Adj\left(A\right)}{\begin{vmatrix}A\end{vmatrix} \\} $$

Where:

$$ Adj \left(A\right) = \begin{bmatrix}d & -b\\ -c & a \\\end{bmatrix} $$

For

$$ A = \begin{bmatrix}a & b\\ c & d \\\end{bmatrix} $$

$$ det A = \begin{vmatrix}a & b \\c & d\end{vmatrix} \\ = ad – bc $$

For inverse of matrix, following condition should be satisfied

  • The matrix should be a square matrix.
  • The determinant of the matrix |A|≠ 0

We can check by the inverse matrix calculator whether the matrix is fulfilling the above conditions or not.

What about if you think of deredining the adjoint and inverse of 3×3 matrix.
If
$$ A= \begin{vmatrix}a & b & c \\d & e & f \\g & h & i\end{vmatrix} $$

Then

$$ Adj A = \begin{vmatrix}M_{11} & M_{12} & M_{13} \\M_{12} & M_{22} & M_{23} \\M_{31} & M_{32} & M_{33}\end{vmatrix}^{t}$$

To determine the inverse of a 3×3 matrix we have to deal with the concept of minors and cofactors.

Minor:

The minor is defined for each and every element of the matrix. The minor of specific element is the determinant obtained after eliminating the row and column containing that element.

Cofactor:

The Cofactor of an element is determined by multiplying the minor with the exponent sum of the row and the column of the specific element.
$$ Cofactor of a_{ij}= (-1)^i+j × minor of a_{ij} $$

For the given matrix A above, we are finding the minor and the cofactor of the matrix.
$$ A= \begin{vmatrix}a & b & c \\d & e & f \\g & h & i\end{vmatrix} $$
$$ M_{11} = \left(-1\right)^{i+j} \begin{vmatrix}e & f \\h & i\end{vmatrix} \\ $$
$$ M_{12} = \left(-1\right)^{i+j} \begin{vmatrix}d & f \\g & i\end{vmatrix} \\ $$
$$ M_{13} = \left(-1\right)^{i+j} \begin{vmatrix}d& e \\g & h\end{vmatrix} \\ $$
$$ M_{21} = \left(-1\right)^{i+j} \begin{vmatrix}b & c \\h & i\end{vmatrix} \\ $$
$$ M_{22} = \left(-1\right)^{i+j} \begin{vmatrix}a & c \\g & i\end{vmatrix} \\ $$
$$ M_{23} = \left(-1\right)^{i+j} \begin{vmatrix}d & e \\g & h\end{vmatrix} \\ $$
$$ M_{31} = \left(-1\right)^{i+j} \begin{vmatrix} e & g \\h & i\end{vmatrix} \\ $$
$$ M_{32} = \left(-1\right)^{i+j} \begin{vmatrix} a & c\\g & i\end{vmatrix} \\ $$
$$ M_{33} = \left(-1\right)^{i+j} \begin{vmatrix} d & e\\g & h\end{vmatrix} \\ $$
$$ \text{Cofactor Matrix} = \begin{vmatrix}M_{11} & M_{12} & M_{13} \\M_{21} & M_{22} & M_{23} \\M_{31} & M_{32} & M_{33}\end{vmatrix} $$
$$ A^{t} = \begin{vmatrix}M_{11} & M_{12} & M_{13} \\M_{21} & M_{22} & M_{23} \\M_{31} & M_{32} & M_{33}\end{vmatrix}^{t}$$
$$ Adj\left(A\right) = \begin{vmatrix}M_{11} & M_{12} & M_{13} \\M_{12} & M_{22} & M_{23} \\M_{31} & M_{32} & M_{33}\end{vmatrix}^{t}$$
The whole calculation of the inverse of 3×3 matrix can be done swiftly by the inverse matrix calculator.

Determinant:

The determinant of a matrix is a single unique representation of the matrix. The determinant of the matrix is equal to the summation of the products of the elements and its cofactors for a particular row and column of a matrix. We can find the determinant of the matrix by the determinant calculator.

Singular Matrix:

A max having the determinant value equal to the Zero is known to be as the singular matrix. For the singular matrix A, |A| = 0, We can’t find the inverse of a singular matrix; this condition applies if we find the inverse of a 3×3 matrix or any other square matrix.

Non-Singular Matrix:

A matrix whose determinant is not equal to the Zero is known as the non-singular matrix. For a non singular matrix |A| ≠ 0, and it is also called the invertible matrix since its inverse can be calculated.

Gauss Jordan Method:

We can implement the Gauss Jordan method by the following operations:
$$ \begin{bmatrix} a&b&c \\ d&e&f \\g&h& i \end{bmatrix}\\ $$
We are going to make the matrix an identity matrix by applying the row operations.
$$ \left[\begin{array}{ccc|ccc}a&b&c&1&0&0\\ d&e&f&0&1&0\\g & h& i&0&0&1 \end{array}\right]\\ $$
We need to implement the row operation to find the inverse of the matrix.
We need to change our matrix into the identity matrix
Then we need to implement the row operation.
The result would be our inverse matrix by the Gauss Jordan Elimination method.
The inverse matrix calculator swiftly finds the inverse of a matrix by the Gauss Jordan Elimination method.
Example:
Calculate the and solve the inverse of a 3×3 matrix by the Gauss Jordan Elimination method:
$$ \begin{bmatrix}1&1&9 \\ 2&5&1\\1&2&7\end{bmatrix}\\ $$
Now find the determinant:
We are going to make the matrix an identity matrix by applying the row operations.
$$ \left[\begin{array}{ccc|ccc}1&1&9&1&0&0\\ 2&5&1&0&1&0\\1&2&7&0&0&1\end{array}\right]\\ $$
Our final Inverse matrix by the Gauss Jordan elimination method is:
$$ Inverse matrix= \begin{bmatrix}3&1&-4 \\ -1.182&-0.182&1.545 \\ -0.091&-0.091&0.273 \\\end{bmatrix} $$
The inverse of matrix calculator can find the inverse of a 3×3 matrix in a matter of seconds.

Working of the Inverse Matrix Calculator:

The inverse matrix calculation is simple to find when using the inverse of the matrix calculator. This can be done in a matter of moments in the most simple and effective manner.
Input:

  • Choose the size of the matrix  from the drop down menu
  • Enter the values and hit the Generate Matrix button
  • Choose the method to solve the inverse matrix
  • Hit the calculate button

Output:
The invertible matrix is easily converted into its inverse matrix by the invertible matrix calculator.

  • The input matrix values are displayed
  • All the steps of calculations are displayed

FAQs:

What is Invertible matrix?

A matrix having an inverse of matrix, it should be non- singular and square in nature.

Can we find the inverse of all matrices?

We aren’t able to find the inverse of all the matrices, only the invertible matrix inverse can be determined by the matrix inverse calculator.

Can I obtain the original matrix after inverting it?

You only need to take the following steps to obtain the original matix by the inverse of a matrix calculator:

  • Enter your inverted matrix.
  • Hit the calculate button to get the inverse of your inverted matrix.
  • It  gives you your original matrix.

Can you invert a singular matrix?

No you can’t invert a singular matrix because when calculating the inverse of a matrix the determinant becomes equal to zero. You can use the inverse matrix calculator to find whether a matrix is singular or not.

Conclusion:

We need to find the inverse of the matrix to find the solution of the linear by the matrix inversion method. The inverse of 3×3 matrix, and inverse of 4×4 matrix is a lengthy procedure and we need the special inverse matrix.

References:

From the source of Wikipedia: Invertible matrix , Properties ,  Other Properties

From the source of embibe.com:Inverse Matrix , Meaning, Formula, Solved Examples