Inverse of a Matrix

The inverse of Matrix is the matrix that on multiplying with the original matrix results in an identity matrix. For any matrix A, its inverse is denoted as A^-1.

Let’s learn about the Matrix Inverse in detail, including its definition, formula, methods on how to find the inverse of a matrix, and examples.

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Matrix Inverse

The inverse of a matrix is another matrix that, when multiplied by the given matrix, yields the multiplicative identity.

For matrix A and its inverse of A^-1, the identity property holds.

A.A^-1 = A^-1A = I

where I is the identity matrix.

Terms Related to Matrix Inverse

The terminology listed below can help you grasp the inverse of a matrix more clearly and easily.

Terms	Definition	Formula/Process	Example with Matrix A
Minor	The minor of an element in a matrix is the determinant of the matrix formed by removing the row and column of that element.	For element aij , remove the ith row and jth column to form a new matrix and find its determinant.	Minor of a11 is the determinant of [Tex]A = \begin{bmatrix}5 & 6\\ 6 & 7\end{bmatrix} [/Tex]
Cofactor	The cofactor of an element is the minor of that element multiplied by (-1)i+j , where i and j are the row and column indices of the element.	Cofactor of aij = (-1)i+j Minor of aij	Cofactor of a11 = (-1)1+1 × Minor of a11 = Minor of a11
Determinant	The determinant of a matrix is calculated as the sum of the products of the elements of any row or column and their respective cofactors.	For a row (or column), sum up the product of each element and its cofactor.	Determinant of A = a11 ​× Cofactor of a11 +a12 × Cofactor of a12​ +a13 × Cofactor of a13​.
Adjoint	The adjoint of a matrix is the transpose of its cofactor matrix.	Create a matrix of cofactors for each element of the original matrix and then transpose it.	Adjoint of A is the transpose of the matrix formed by the cofactors of all elements in A.

Singular Matrix

A matrix whose value of the determinant is zero is called a singular matrix, i.e. any matrix A is called a singular matrix if |A| = 0. Inverse of a singular matrix does not exist.

Non-Singular Matrix

A matrix whose value of the determinant is non-zero is called a non-singular matrix, i.e. any matrix A is called a non-singular matrix if |A| ≠ 0. Inverse of a non-singular matrix exists.

Identity Matrix

A square matrix in which all the elements are zero except for the principal diagonal elements is called the identity matrix. It is represented using I. It is the identity element of the matrix as for any matrix A,

A×I = A

An example of an Identity matrix is,

I_3×3 =[Tex] \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{bmatrix}[/Tex]

This is an identity matrix of order 3×3.\

Read More :

• Identity Matrix

How to Find Inverse of Matrix?

There are Two-ways to find the Inverse of a matrix in mathematics:

• Using Matrix Formula
• Using Inverse Matrix Methods

Inverse of a Matrix Formula

The inverse of matrix A, that is A^-1 is calculated using the inverse of matrix formula, which involves dividing the adjoint of a matrix by its determinant.

[Tex]A^{-1}=\frac{\text{Adj A}}{|A|}   [/Tex]

where,

• adj A = adjoint of the matrix A, and 
• |A| = determinant of the matrix A.

Note: This formula only works on Square matrices.

To find inverse of matrix using inverse of a matrix formula, follow these steps.

Step 1: Determine the minors of all A elements.

Step 2: Next, compute the cofactors of all elements and build the cofactor matrix by substituting the elements of A with their respective cofactors.

Step 3: Take the transpose of A’s cofactor matrix to find its adjoint (written as adj A).

Step 4: Multiply adj A by the reciprocal of the determinant of A.

Now, for any non-singular square matrix A,

A^-1 = 1 / |A| × Adj (A)

Example: Find the inverse of the matrix [Tex]A=\left[\begin{array}{ccc}4 & 3 & 8\\6 & 2 & 5\\1 & 5 & 9\end{array}\right]            [/Tex] using the formula.

We have, [Tex]A=\left[\begin{array}{ccc}4 & 3 & 8\\6 & 2 & 5\\1 & 5 & 9\end{array}\right] [/Tex]

Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose.

adj A = [Tex]\left[\begin{array}{ccc}-7 & -49 & 28\\13 & 28 & -17\\-1 & 28 & -10\end{array}\right] [/Tex]

Find the value of determinant of the matrix.

|A| = 4(18–25) – 3(54–5) + 8(30–2)

⇒ |A| = 49

So, the inverse of the matrix is,

A^–1 = [Tex]\frac{1}{49}\left[\begin{array}{ccc}-7 & -49 & 28\\13 & 28 & -17\\-1 & 28 & -10\end{array}\right] [/Tex]

⇒ A^–1 = [Tex]\left[\begin{array}{ccc}- \frac{1}{7} & \frac{13}{49} & – \frac{1}{49}\\-1 & \frac{4}{7} & \frac{4}{7}\\\frac{4}{7} & – \frac{17}{49} & – \frac{10}{49}\end{array}\right] [/Tex]

Inverse Matrix Method

There are two Inverse matrix methods to find matrix inverse:

1. Determinant Method
2. Elementary Transformation Method

Method 1: Determinant Method

The most important method for finding the matrix inverse is using a determinant.

The inverse matrix is also found using the following equation:

A^-1= adj(A) / det(A)

where,

• adj(A) is the adjoint of a matrix A, and
• det(A) is the determinant of a matrix A.

For finding the adjoint of a matrix A the cofactor matrix of A is required. Then adjoint (A) is the transpose of the Cofactor matrix of A i.e.,

adj (A) = [C_ij]^T

• For the cofactor of a matrix i.e., C_ij, we can use the following formula:

C_ij = (-1)^i+j det (M_ij)

where M_ij refers to the (i, j)^th minor matrix when i^th row and j^th column is removed.

Method 2: Elementary Transformation Method

Follow the steps below to find an Inverse matrix by elementary transformation method.

Step 1 : Write the given matrix as A = IA, where I is the identity matrix of the order same as A.

Step 2 : Use the sequence of either row operations or column operations till the identity matrix is achieved on the LHS also use similar elementary operations on the RHS such that we get I = BA. Thus, the matrix B on RHS is the inverse of matrix A.

Step 3 : Make sure we either use Row Operation or Column Operation while performing elementary operations.

We can easily find the inverse of the 2 × 2 Matrix using the elementary operation. Let’s understand this with the help of an example.

Example: Find the inverse of the 2 × 2, A =  [Tex]     [/Tex] using the elementary operation.

Solution:

Given:

A = IA

[Tex]\begin{bmatrix}2 & 1\\ 1 & 2\end{bmatrix}~=~\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}~×~\begin{bmatrix}2 & 1\\ 1 & 2\end{bmatrix} [/Tex]

Now, R₁ ⇢ R₁/2

[Tex]\begin{bmatrix}1 & 1/2\\ 1 & 2\end{bmatrix}~=~\begin{bmatrix}1/2 & 0\\ 0 & 1\end{bmatrix}~×~A [/Tex]

R₂ ⇢ R₂ – R₁

[Tex]\begin{bmatrix}1 & 1/2\\ 0 & 3/2\end{bmatrix}~=~\begin{bmatrix}1/2 & 0\\ -1/2 & 1\end{bmatrix}~×~A [/Tex]

R₂ ⇢ R₂ × 2/3

[Tex]\begin{bmatrix}1 & 1/2\\ 0 & 1\end{bmatrix}~=~\begin{bmatrix}1/2 & 0\\-1/3 & 2/3\end{bmatrix}~×~A [/Tex]

R₁ ⇢ R₁ – R₂/2

[Tex]\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}~=~\begin{bmatrix}2/3 & -1/6\\ -1/3 & 2/3\end{bmatrix}~×~A [/Tex]

Thus, the inverse of the matrix A = [Tex] \begin{bmatrix}2 & 1\\ 1 & 2\end{bmatrix}   [/Tex] is

A^-1 = [Tex]\begin{bmatrix}2/3 & -1/6\\ -1/3 & 2/3\end{bmatrix} [/Tex]

Inverse of 2×2 Matrix Example

Inverse of the 2×2 matrix can also be calculated using the shortcut method apart from the method discussed above. Let’s consider an example to understand the shortcut method to calculate the inverse of 2 × 2 Matrix.

For given matrix A = [Tex]\begin{bmatrix}a & b\\ c & d\end{bmatrix} [/Tex]

We know, |A| = (ad – bc)

and adj A = [Tex]\begin{bmatrix}d & -b\\ -c & a\end{bmatrix} [/Tex]

then using the formula for inverse

A^-1 =  (1 / |A|) × Adj A

⇒ A^-1 = [Tex][1 / (ad – bc)] × \begin{bmatrix}d & -b\\ -c & a\end{bmatrix} [/Tex]

Thus, the inverse of the 2 × 2 matrix is calculated.

Inverse of 3X3 Matrix Example

Let us take any 3×3 Matrix A = [Tex]\begin{bmatrix}a & b & c\\ l & m & n\\ p & q & r\end{bmatrix} [/Tex]

The inverse of 3×3 matrix is calculated using the inverse matrix formula, 

A^-1 = (1 / |A|) × Adj A

Determinant of Inverse Matrix

Determinant of inverse matrix is the reciprocal of the determinant of the original matrix. i.e., 

det(A^-1) = 1 / det(A)

The proof of the above statement is discussed below:

det(A × B) = det (A) × det(B)  (already know)

⇒ A × A^-1 = I  (by Inverse matrix property)

⇒ det(A × A^-1) = det(I)

⇒ det(A) × det(A^-1) = det(I)     [ but, det(I) = 1]

⇒ det(A) × det(A^-1) = 1

⇒ det(A^-1) = 1 / det(A)

Hence, Proved.

Properties of Inverse of Matrix

Inverse matrix has the following properties:

• For any non-singular matrix A, (A^-1)^-1 = A
• For any two non-singular matrices A and B, (AB)^-1 = B^-1A^-1
• Inverse of a non-singular matrix exists, for a singular matrix, the inverse does not exist.
• For any nonsingular A, (A^T)^-1 = (A^-1)^T

Related :

• Invertible Matrix
• Matrices: Properties and Formulas
• Mathematical Operation on Matrices
• Determinant of Matrix
• How to find the Determinant of Matrix?

Matrix Inverse Solved Examples

Let’s solve some example questions on Inverse of Matrix.

Example 1: Find the inverse of the matrix [Tex]\bold{A=\left[\begin{array}{ccc}2 & 3 & 1\\1 & 1 & 2\\2 & 3 & 4\end{array}\right]      }      [/Tex] using the formula.

Solution:

We have,

[Tex]A=\left[\begin{array}{ccc}2 & 3 & 1\\1 & 1 & 2\\2 & 3 & 4\end{array}\right] [/Tex]

Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose.

adj A = [Tex]\left[\begin{array}{ccc}-2 & -9 & 5\\0 & 6 & -3\\1 & 0 & -1\end{array}\right] [/Tex]

Find the value of determinant of the matrix.

|A| = 2(4–6) – 3(4–4) + 1(3–2)

= –3

So, the inverse of the matrix is,

A^–1 = [Tex]\frac{1}{-3}\left[\begin{array}{ccc}-2 & -9 & 5\\0 & 6 & -3\\1 & 0 & -1\end{array}\right] [/Tex]

      = [Tex]\left[\begin{array}{ccc}\frac{2}{3} & 3 & – \frac{5}{3}\\0 & -2 & 1\\- \frac{1}{3} & 0 & \frac{1}{3}\end{array}\right] [/Tex]

Example 2: Find the inverse of the matrix A=\bold{ using the formula.}[Tex]\left[\begin{array}{ccc}6 & 2 & 3\\0 & 0 & 4\\2 & 0 & 0\end{array}\right]       [/Tex]

Solution:

We have,

A=[Tex]\left[\begin{array}{ccc}6 & 2 & 3\\0 & 0 & 4\\2 & 0 & 0\end{array}\right] [/Tex]

Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose.

adj A = [Tex]\left[\begin{array}{ccc}0 & 0 & 8\\8 & -6 & -24\\0 & 4 & 0\end{array}\right] [/Tex]

Find the value of determinant of the matrix.

|A| = 6(0–4) – 2(0–8) + 3(0–0)

     = 16

So, the inverse of the matrix is,

A^–1 = [Tex]\frac{1}{16}\left[\begin{array}{ccc}0 & 0 & 8\\8 & -6 & -24\\0 & 4 & 0\end{array}\right] [/Tex]

      = [Tex]\left[\begin{array}{ccc}0 & 0 & \frac{1}{2}\\\frac{1}{2} & – \frac{3}{8} & – \frac{3}{2}\\0 & \frac{1}{4} & 0\end{array}\right] [/Tex]

Example 3: Find the inverse of the matrix A=[Tex]\bold{\left[\begin{array}{ccc}1 & 2 & 3\\0 & 1 & 4\\0 & 0 & 1\end{array}\right]      }      [/Tex] using the formula.

Solution:

We have,

A=[Tex]\left[\begin{array}{ccc}1 & 2 & 3\\0 & 1 & 4\\0 & 0 & 1\end{array}\right] [/Tex]

Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose.

adj A = [Tex]\left[\begin{array}{ccc}1 & -2 & 5\\0 & 1 & -4\\0 & 0 & 1\end{array}\right] [/Tex]

Find the value of determinant of the matrix.

|A| = 1(1–0) – 2(0–0) + 3(0–0)

= 1

So, the inverse of the matrix is,

A^–1 = [Tex]\frac{1}{1}\left[\begin{array}{ccc}1 & -2 & 5\\0 & 1 & -4\\0 & 0 & 1\end{array}\right] [/Tex]

= [Tex]\left[\begin{array}{ccc}1 & -2 & 5\\0 & 1 & -4\\0 & 0 & 1\end{array}\right] [/Tex]

Example 4: Find the inverse of the matrix A=[Tex]\bold{\left[\begin{array}{ccc}1 & 2 & 3\\2 & 1 & 4\\3 & 4 & 1\end{array}\right]      }      [/Tex] using the formula.

Solution:

We have,

A=[Tex]\left[\begin{array}{ccc}1 & 2 & 3\\2 & 1 & 4\\3 & 4 & 1\end{array}\right] [/Tex]

Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose.

adj A = [Tex]\left[\begin{array}{ccc}-15 & 10 & 5\\10 & -8 & 2\\5 & 2 & -3\end{array}\right] [/Tex]

Find the value of determinant of the matrix.

|A| = 1(1–16) – 2(2–12) + 3(8–3)

= 20

So, the inverse of the matrix is,

A^–1 = [Tex]\frac{1}{20}\left[\begin{array}{ccc}-15 & 10 & 5\\10 & -8 & 2\\5 & 2 & -3\end{array}\right] [/Tex]

      = [Tex]\left[\begin{array}{ccc}- \frac{3}{4} & \frac{1}{2} & \frac{1}{4}\\\frac{1}{2} & – \frac{2}{5} & \frac{1}{10}\\\frac{1}{4} & \frac{1}{10} & – \frac{3}{20}\end{array}\right] [/Tex]

Frequently Asked Questions on Inverse of Matrix

What is Inverse of Matrix?

Reciprocal of a matrix is called the Inverse of a matrix. Only square matrices with non-zero determinants are invertible. Suppose for any square matrix A with inverse matrix B their product is always an identity matrix (I) of the same order.

[A]×[B] = [I]

What is Matrix?

Matrix is a rectangular array of numbers that are divided into a defined number of rows and columns. The number of rows and columns in a matrix is referred to as its dimension or order.

What is the Inverse of 2×2 Matrix?

For any matrix A or order 3×3 its inverse is found using the formula,

A^-1 = (1 / |A|) × Adj A

What is the Inverse of 3×3 Matrix?

The inverse of any square 3×3 matrix (say A) is the matrix of the same order denoted by A^-1 such that their product is an Identity matrix of order 3×3.

[A]_3×3 × [A^-1]_3×3 = [I]_3×3

Are Adjoint and Inverse of Matrix the same?

No, the adjoint of a matrix and the inverse of a matrix are not the same. 

How to use the Inverse of Matrix?

The inverse of a matrix is used for solving algebraic expressions in matrix form. For example, to solve AX = B, where A is the coefficient matrix, X is the variable matrix and B is the constant matrix. Here the variable matrix is found using the inverse operation as,

X = A^-1B

What are Invertible Matrices?

The matrices whose inverse exist are called invertible. Invertible matrices are matrices that have a non-zero determinant.

Why does Inverse of 2 × 3 Matrix not exist?

The inverse of only a square matrix exists. As the 2 × 3 matrix is not a square matrix but rather a rectangular matrix thus, its inverse does not exist.

Similarly, the 2 × 1 matrix is also not a square matrix but rather a rectangular matrix thus, its inverse does not exist.

What is Inverse of Identity Matrix?

The inverse of an identity matrix is the identity matrix itself. This is because the identity matrix, denoted as I (or I_n for an n ×n matrix), is the only matrix for which every element along the main diagonal is 1 and all other elements are 0. When we multiply an identity matrix by itself (or its inverse), we get the identity matrix again.