Inverse Matrix Calculator is a mathematical tool that does all of the time-consuming and difficult calculations required to discover the Inverse of a given matrix in seconds. To find the inverse of a Matrix, there are several methods and shortcuts. To execute the necessary function, an Inverse Matrix Calculator often employs the Gauss-Jordan (also known as Elementary Row Operations) technique and the Adjoint method. When utilising the Inverse Matrix Calculator, you don’t have to conduct all of the calculation steps yourself; simply enter the appropriate numbers into the calculator, and it will do the rest in a matter of seconds. Let’s look at how to utilise an inverse matrix calculator using the instructions in this post. Continue reading to learn more.

Inverse Matrix Calculator

In this article, we will be discussing in detail the Inverse Matrix Calculator, its features, how it works, examples, etc. To understand the functionality of this tool, first, we need to understand all the essential teams associated with it. Like you should be clear about what is a matrix & its inverse, components of a matrix, operations and mathematical methods used for computing the Inverse Matrix. So, without wasting any time, scroll to get all vital details about the Inverse Matrix Calculator available on this page.

What is Matrix and Inverse Matrix?

A Matrix is nothing but an arrangement of numbers, symbols, expressions or other mathematical objects in the form of rows and columns. The numbers, symbols, or expressions which form the matrix are called Matrix entries or Matrix elements. The horizontal and vertical lines of elements in matrices are termed as rows and columns, respectively. ‘Matrices’ is the plural form of a matrix. Mathematical operations such as addition, subtraction and multiplication can be performed on the matrices.

However, we cannot divide a matrix by the other; for division, we use the concept of Inverse Matrix, which is discussed in the latter portion of the page. The significance of matrices lies in their numerous applications in Statistics, Economics, Engineering, Physics and other branches of Mathematics.

A matrix can be real or complex based on the nature of its elements. For example, the following is a real matrix that contains all the natural numbers as its entries:

$M=\begin{array}{c}1–952.5–6.810\end{array}$

Size of a Matrix

A total number of rows and columns defines the size of a matrix. There is no limit to the numbers of rows and columns a matrix (in the standard sense) can have as long as they’re positive integers. A matrix with m rows and n columns is termed an m × n matrix, or m-by-n matrix, while m and n are called its dimensions. For example, the matrix M above is 3 × 2 matrices, i.e. it contains 3 rows and 2 columns. A matrix having the same number of rows and columns is called a Square Matrix (m=n).

Inverse of a Matrix

The inverse of matrix M is M^-1 only when:

M × M^-1 = M^-1 × M = I

‘I’ represents an Identity Matrix that is equivalent of the number “1”. A 3×3 Identity Matrix is expressed as:

$I=\begin{array}{c}100010001\end{array}$

The Identity Matrix is always square in nature and has 1s on the diagonal and 0s everywhere else. It can be 2×2 in size, or 3×3, 4×4, and so on…

For better understanding, let us find how to calculate the inverse of a 2×2 matrix:

$\mathrm{Inverse} \mathrm{of} \mathrm{matrix} M, \left[\begin{array}{c}\mathrm{abcd}\end{array}\right]=\frac{1}{\mathrm{ad}–\mathrm{bc}} \left[\begin{array}{c}d–b–\mathrm{ca}\end{array}\right]$

$\mathrm{where}, \frac{1}{\mathrm{ad}–\mathrm{bc}}\mathrm{represents} \mathrm{the} \mathrm{determinant} \mathrm{of} M.$

$$\begin{gathered} \mathit{Example}\mathbf{:}\text{ \\ Inverse of matrix 4276 = 14×6–7×26–7–24 \\ =1106–7–24 \\ =0.6–0.7–0.20.4} \end{gathered}$$

$$\begin{gathered} \text{ \\ Now we will check for the Identity (A × A–1 = I) by multiplying the \\ matrix by its Inverse we just calculated} \end{gathered}$$

$$\begin{gathered} \text{ \\ =4276 0.6–0.7–0.20.4 \\ =4×0.6+7×-0.24×-0.7+7×0.42×0.6+6×-0.22×-0.7+6×0.4 \\ =2.4–1.4–2.8+2.81.2–1.2–1.4+2.4 \\ =1001 \\ } \end{gathered}$$

Why do we need to find an Inverse of a Matrix? 

We cannot perform the division operation on matrices. So to divide a matrix by another matrix, we can multiply by an inverse, which gives us the same result as intended to obtain after division. 

What is Inverse Matrix Calculator?

As we have briefly explained above in the article, it is a mathematical tool that computes the inverse of a matrix within seconds, saving us valuable time and hectic calculations. Now you all know very well about Matrices and their inverse, we can focus on the functionality of the Inverse Matrix Calculator. Keep in mind the inverse can be computed for square matrices only. 

How to Use the Inverse Matrix Calculator?

Please follow the steps below to find the inverse matrix using an online inverse matrix calculator:

• Step 1: First of all, choose the size of the Square Matrix (for instance, 4, 5, etc.)
• Step 2: The matrix with input boxes of entered size will be displayed in front of you. Enter the elements of the matrix in their respective places.
• Step 3: Press on the “Calculate” button to find the resultant inverse matrix. The Inverse Matrix Calculator will display the processed result with a detailed solution on the screen in seconds. 

Note: The Matrix calculator will be embedded soon.

Continue calculation & Recalculate:

• One more feature of the Inverse Matrix Calculator you can use is to calculate or recalculate the inverse of the matrix implementing another method. Select the corresponding button and choose the new method to be applied.
• If you have chosen to continue the calculation, the method will be applied to the result matrix. If chosen recalculate, the method will be used to the original matrix.

Methods to Find Inverse of a Matrix

The methods to find inverse of a matrix are explained below:

1. Gauss-Jordan Elimination Method

A method of solving a linear system of equations. This is done by transforming the system’s augmented matrix into reduced row-echelon form by means of row operations.
Use Gauss-Jordan elimination to transform [ A | I ] into [ I | A^-1 ].

$$\begin{gathered} Example: The following steps result in {\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]}^{–1} \\ \left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|\right] \to \left[\begin{array}{cc}1& 2\\ 0& –2\end{array}\left|\begin{array}{cc}1& 0\\ –3& 1\end{array}\right|\right] \\ \to \left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\left|\begin{array}{cc}1& 0\\ 3/2& –1/2\end{array}\right|\right] \\ \to \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\left|\begin{array}{cc}–2& 1\\ 3/2& –1/2\end{array}\right|\right] \\ so we see that {\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]}^{–1}= \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\left|\begin{array}{cc}–2& 1\\ 3/2& –1/2\end{array}\right|\right] \end{gathered}$$

2. Adjoint Method

$A^{–1} =\frac{1}{det A} (adjoint of A) or A^{–1} =\frac{1}{det A} (cofactor matrix of A{)}^T$

$$\begin{gathered} Example: The following steps result in A^{–1} for A = \left[\begin{array}{ccc}1& 2& 3\\ 1& 4& 5\\ 0& 1& 6\end{array}\right] \\ The cofactor matrix for A is \left[\begin{array}{ccc}24& 5& –4\\ –12& 3& 2\\ –2& –5& 4\end{array}\right] , so the adjoint is \\ \left[\begin{array}{ccc}24& –12& –2\\ 5& 3& –5\\ –4& 2& 4\end{array}\right]. Since det A = 22, we get \\ {A}^{–1} = \frac{1}{22} \left[\begin{array}{ccc}24& –12& –2\\ 5& 3& –5\\ –4& 2& 4\end{array}\right] = \left[\begin{array}{ccc}12/11& –6/11& –1/11\\ 5/22& 3/22& –5/22\\ –2/11& 1/11& 2/11\end{array}\right] \end{gathered}$$

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FAQs Related to Inverse Matrix Calculator

Q. How do you find the inverse of a 3×3 matrix?
A. Two methods can be employed to find the inverse of a 3×3 matrix, namely, the Gauss-Jordan Elimination Method and the Adjoint Method. These methods have been explained properly on this page.

Q. Can a matrix have multiple inverses?
A square matrix cannot have more than one multiplicative inverse. But if talk left or right inverse of a non-square matrix, it can have many infinite inverses. 

Q. Can you find the inverse of a non-square matrix?
A. No, Non-square matrices (m-by-n matrices for which m ≠ n) do not have an inverse. If A has rank m, then it has a right inverse: an n-by-m matrix B such that AB = I. A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0.

Q. How do you find the inverse of a 2×2 matrix?
A. The shortcut method of finding the inverse of a 2×2 matrix is well discussed in details on this page.

Q. Which matrix has no inverse?
A. A matrix whose determinant is 0 is called a singular matrix. A singular matrix does not have an inverse.

Q. What are the properties of an inverse matrix?
If A is nonsingular, then so is A^-1 and (A^-1) ^-1 = A.
If A and B are nonsingular matrices, then AB is nonsingular and. (AB) ^-1 = B^-1A^-1 -1
If A is nonsingular then. (A^T) ^-1 = (A ^-1)^T
If A and B are matrices with. AB = I_n then A and B are inverses of each other.

This is all you need to learn in the Inverse matrix Calculator. Apart from that, you can explore numerous other topics explained on embibe.com.