## Introduction to an Inverse Matrix Calculator:

Inverse matrix calculator is a digital tool that is designed to **calculate the inverse** of a given matrix in a run of time. Our tool has advanced algorithms that help you to get the exact solution of inverse matrix problems quickly without any mistakes.

The matrix inverse calculator is a beneficial tool for students to learn linear algebra and its complex calculations like the inverse matrix in a simplified process without going anywhere.

## What is an Inverse Matrix?

An inverse matrix is one of the important operations used in the matrix in which a **square matrix** A exists if there is another square matrix, denoted as A^−1 present, such that A is multiplied by inverse A as A^{−1}. The result is the identity matrix I and it is known as the inverse of A. Mathematically,

if A is an n×n matrix and I is the n×n identity matrix, then

$$ A \times A^{-1} \;=\; I $$

For a square matrix A the inverse of the matrix A exists if it is non-singular (its determinant is not zero). The matrix rows (or columns) are linearly independent.

## How to Calculate the Inverse of a Matrix

For **calculating the matrix inverse**, the square matrix A, and its A^{-1} must exist. There are some methods often used by the inverse matrix calculator such as the row operations, Gaussian elimination, or the adjugate matrix for calculations.

Let us see how to find the inverse of a matrix A.

**Step 1**:

First, the inverse calculator matrix finds the determinant of a square matrix A. Check whether the matrix is nonsingular or not. if it is not nonsingular then you cannot find the inverse of a given matrix.

**Step 2**:

The inverse of a matrix calculator expands the given **matrix with its cofactors** after opening the determinant.

**Step 3**:

The solution of the cofactor value becomes the determinant matrix into a new matrix

**Step 4**:

The inverse of matrix calculator finds the adjoint of a new matrix that is obtained after finding the determinant in which the right side of the diagonal sign has changed and the left side of the diagonal has interchanged its position.

**Step 5**:

The inverse matrix calculator checks the result of the inverse of a square matrix by **multiplying the square matrix** A and A^{-1}. If the solution of A and A^{-1} is equal to identify matrix it means your solution is correct with the help of that formula as

$$ AA^{-1} \;=\; I $$

## Solved Example of Inverse Matrix

The matrix inverse calculator can help you solve the inverse matrix problem but you should know how to do **manual calculation** as well. So an example is given,

**Example**:

$$ \biggr[\begin{matrix} 1 & 5 & 2 \\ 0 & -1 & 2 \\ 0 & 0 & 1 \\ \end{matrix} \biggr] $$

**Solution**:

$$ AA^{-1} \;=\; A^{-1} A \;=\; I $$

$$ A_{11} \;=\; \biggr|\begin{matrix} -1 & 2 \\ 0 & 1 \\ \end{matrix} \biggr| \;=\; -1 $$

$$ A_{21} \;=\; -\biggr|\begin{matrix} 0 & 2 \\ 0 & 1 \\ \end{matrix} \biggr| \;=\; 0 $$

$$ A_{13} \;=\; \biggr|\begin{matrix} 0 & -1 \\ 0 & 0 \\ \end{matrix} \biggr| \;=\; 0 $$

$$ A_{21} \;=\; -\biggr|\begin{matrix} 5 & 2 \\ 0 & 1 \\ \end{matrix} \biggr| \;=\; -5 $$

$$ A_{22} \;=\; \biggr|\begin{matrix} 1 & 2 \\ 0 & 1 \\ \end{matrix} \biggr| \;=\; 1 $$

$$ A_{23} \;=\; -\biggr|\begin{matrix} 1 & 5 \\ 0 & 0 \\ \end{matrix} \biggr| \;=\; 0 $$

$$ A_{31} \;=\; \biggr|\begin{matrix} 5 & 2 \\ -1 & 2 \\ \end{matrix} \biggr| \;=\; 12 $$

$$ A_{32} \;=\; -\biggr|\begin{matrix} 1 & 2 \\ 0 & 2 \\ \end{matrix} \biggr| \;=\; -2 $$

$$ A_{33} \;=\; -\biggr|\begin{matrix} 1 & 5 \\ 0 & -1 \\ \end{matrix} \biggr| \;=\; -1 $$

After finding the determinant of the given matrix

$$ \biggr[\begin{matrix} -1 & -5 & 12 \\ 0 & 1 & -2 \\ 0 & 0 & -1 \\ \end{matrix} \biggr] $$

By the formula of inverse matrix,

$$ A^{-1} \;=\; \frac{adj(A)}{det(A)} \;=\; \biggr[\begin{matrix} 1 & 5 & -12 \\ 0 & -1 & 2 \\ 0 & 0 & 1 \\ \end{matrix} \biggr] $$

## How to Use Inverse Matrix Calculator?

The inverse calculator matrix has an easy-to-use design that enables you to use it to solve the inverse of given matrix questions easily. You must follow some simple steps to avoid trouble during the calculation. These simple steps are:

- Choose the size of a square matrix A in the input box.
- Enter the value of A square matrix in the input box.
- Recheck your input value before hitting the calculate button to start the calculation process in the inverse of a matrix calculator.
- Click on the “
**Calculate**” button to get the desired result for the inverse matrix problem. - If you are trying out our inverse of matrix calculator to check its solution accuracy, use the load example.
- Click on the “Recalculate” button to get a new page to determine more problems related to the inverse matrix.

## Outcome of Matrix Inverse Calculator:

The inverse calculator matrix gives you the solution to a given values matrix inverse problem when you give input to it. It provides you with the result of the inverse matrix in a step-wise process. It may contain:

The result option provides you with a **solution of the inverse matrix** from the given square matrix.

The possible steps option of the inverse matrix calculator provides you with the solution of the inverse matrix where all calculation steps are present.

## Advantages of Inverse Calculator Matrix:

The inverse of a matrix calculator gives you tons of advantages while you are using it to calculate inverse matrix problem solutions. These advantages are:

- Our calculator
**saves the time**and effort that you consume during complex calculations of the inverse of a given matrix solution. - The inverse of matrix calculator is a free-of-cost tool that provides you with a solution for inverse matrix questions.
- It is an adaptive tool that can be used anywhere in the world with the help of the internet for the solution of matrix inverse questions.
- You can use this matrix inverse calculator for practice to get in-depth knowledge about this concept.
- It is a trustworthy tool that provides you with precise solutions as per your input value whenever you use it to find the Inverse Matrix problem.
- The inverse Matrix Calculator provides an accurate solution whenever you use it for the evaluation of linear algebra concepts like inverse matrix questions.