Inverse of Matrix
Inverse of Matrix for a matrix A is A^{1}. The inverse of a 2 x 2 matrix can be calculated using a simple formula. Further, to find the inverse of a 3 x 3 matrix, we need to know about the determinant and adjoint of the matrix. The inverse of matrix is another matrix, which on multiplying with the given matrix gives the multiplicative identity.
The inverse of matrix is used of find the solution of linear equations through the matrix inversion method. Here let us learn about the formula, methods, and the terms related to inverse of matrix.
What is Inverse of Matrix?
The inverse of matrix is another matrix, which on multiplying with the given matrix gives the multiplicative identity. For a matrix A, its inverse is A^{1}, and A.A^{1 } = I. Let us check for the inverse of matrix, for a matrix of order 2 x 2, the general formula for the inverse of matrix is equal to the adjoint of a matrix divided by the determinant of a matrix.
A = \(\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\)
A^{1} = \(\dfrac{1}{ad  bc}\left(\begin{matrix}d&b\\c&a\end{matrix}\right)\)
A^{1} = \(\dfrac{1}{A}\).Adj A
The inverse of matrix exists only if the determinant of the matrix is a nonzero value. The matrix whose determinant is nonzero and for which the inverse matrix can be calculated is called an invertible matrix.
Terms Related to Inverse of Matrix
The following terms below are helpful for more clear understanding and easy calculation of the inverse of matrix.
Minor: The minor is defined for every element of a matrix. The minor of a particular element is the determinant obtained after eliminating the row and column containing this element. For a matrix A = \(\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\), the minor of the element \(a_{11}\) is:
Minor of \(a_{11}\) = \(\left\begin{matrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{matrix}\right\)
Cofactor: The cofactor of an element is calculated by multiplying the minor with 1 to the exponent of the order representation of that element.
Cofactor of \(a_{ij}\) = (1)^{i + j}× minor of \(a_{ij}\).
Determinant: The determinant of a matrix is the single unique value representation of a matrix. The determinant of the matrix can be calculated with reference to any row or column of the given matrix. The determinant of the matrix is equal to the summation of product of the elements and its cofactors, of a particular row or column of the matrix.
Singular Matrix: A matrix having a determinant value of zero is referred to as a singular matrix. For a singular matrix A, A = 0. The inverse of a singular matrix does not exist.
NonSingular Matrix: A matrix whose determinant value is not equal to zero is referred to as a nonsingular matrix. For a nonsingular matrix A ≠0. A nonsingular matrix is called an invertible matrix since its inverse can be calculated.
Adjoint of a Matrix: The adjoint of a matrix is the transpose of the cofactor element matrix of the given matrix.
Rule For Row and Column Operations of a Determinant: The following rules are helpful to perform the row and column operations on determinants.
 The value of the determinant remains unchanged, if the rows and columns are interchanged.
 The sign of the determinant changes, if any two rows or (two columns) are interchanged.
 If any two rows or columns of a matrix are equal, then the value of the determinant is zero.
 If every element of a particular row or column is multiplied by a constant, then the value of the determinant also gets multiplied by the constant.
 If the elements of a row or a column are expressed as a sum of elements, then the determinant can be expressed as a sum of determinants.
 If the elements of a row or column are added or subtracted with the corresponding multiples of elements of another row or column, then the value of the determinant remains unchanged.
Determinant of a Matrix
The determinant of a matrix is a single numeric value that summarizes all the elements of the matrix. The determinant of a matrix is calculated by the summation of the product of the elements and its cofactor of any one row or column of a matrix.
Determinant of a 2 x 2 Matrix
The determinant of a order 2 x 2 matrix is equal to the difference of the product of the elements of the first diagonal and the second diagonal. The determinant of a matrix A = \(\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\) is A = ad  bc.
Determinant of a 3 x 3 Matrix
The determinant of a 3 x 3 matrix can be calculated in 6 different ways. Along three rows (\(R_1\), \(R_2\), \(R_2\)), and along there different columns (\(C_1\), \(C_2\), \(C_3\)). For a matrix A = \(\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\), let us find the determinant along the first row \(R_1\). Multiply the respective elements of the row with its cofactor and find the summation of it.
A = \(a_{11}(1)^{1 + 1} \left\begin{matrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{matrix}\right + a_{12}(1)^{1 + 2} \left\begin{matrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{matrix}\right + a_{13}(1)^{1 + 3} \left\begin{matrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{matrix}\right\)
Adjoint of a Matrix
The adjoint of a matrix is the transpose of the cofactors matrix of the elements of the given matrix. There are three simple steps to form the adjoint of a matrix. First, we need to find the cofactors of each of the elements of the given matrix. Secondly, we need to form the matrix of the cofactor of the elements of the given matrix. Finally, we can form the adjoint of the matrix by taking a transpose of the cofactors elements matrix.
Adjoint of a 2 x 2 Matrix
The adjoint of a 2 x 2 matrix of the form A = \(\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\) is adj A = \(\left(\begin{matrix}d&b\\c&a\end{matrix}\right)\).
Adjoint of a 3 x 3 Matrix
The adjoint of a 3 x 3 matrix can be obtained by following two simple steps. First we need to find the cofactor matrix of the given matrix, and then the transpose of this cofactor matri is taken to obtain the adjoint of a matrix. For a matrix of the form A = \(\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\), the cofactor matrix A = \(\begin{pmatrix} A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}\). Further we need to take the transpose of this cofactor matrix to obtain the adjoing of the matrix.
Adj A = Transpose of Cofactor Matrix = Transpose of \(\begin{pmatrix} A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}\) =\(\begin{pmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\end{pmatrix}\)
Methods to Find Inverse of Matrix
The inverse of matrix can be found using two methods. The inverse of a matrix can be calculated through elementary operations and through the use of an adjoint of a matrix. The elementary operations on a matrix can be performed through a row or column transformations. And the inverse of a matrix can be calculated through the use of the determinant and the adjoint of the matrix. Let us check each of the methods described below.
Also for performing the inverse of matrix through elementary column operations we use the matrix X and the second matrix B on the righthand side of the equation.
Elementary Row Operations
For calculating the inverse of matrix through elementary row operations, let us consider three square matrices X, A, and B respectively. The matrix equation is X = AB. For performing the elementary row operations we use the matrix X and the first matrix A on the righthand side of the equation.
Elementary Column Operations
For calculating the inverse of matrix through elementary column operations, let us consider three square matrices X, A, and B respectively. The matrix equation is X = AB. For performing the elementary column operations we use the matrix X and the second matrix A on the righthand side of the equation.
The inverse of matrix can be computed by dividing the adjoint of a matrix by the determinant of the matrix. For a matrix A, its inverse A^{1} = \(\dfrac{1}{A}\).Adj A.
A = \(\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\)
A = \(a_{11}(1)^{1 + 1} \left\begin{matrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{matrix}\right + a_{12}(1)^{1 + 2} \left\begin{matrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{matrix}\right + a_{13}(1)^{1 + 3} \left\begin{matrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{matrix}\right\)
Adj A = Transpose of Cofactor Matrix = Transpose of \(\begin{pmatrix} A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}\) =\(\begin{pmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\end{pmatrix}\)
A^{1} = \(\dfrac{1}{A}\). \(\begin{pmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\end{pmatrix}\)
Related Topics
The following related links are helpful in the better understanding of the inverse of matrix.
Important Points
The following points are helpful to understand more clearly the idea of the inverse of matrix.
 The inverse of a square matrix if exists, is unique.
 If A and B are two invertible matrices of the same order then (AB)^{1} = B^{1}A^{1}.
 The inverse of a square matrix A exists, only if its determinant is a nonzero value, A ≠0.
 The elements of a row or column, if multiplied with the cofactor elements of any other row or column, then the answer is zero.
 The determinant of the product of two matrices is equal to the product of the determinants of the two individual matrices. AB = A.B
Solved Examples on Inverse of Matrix

Example 1: Find the inverse of matrix A = \(\left(\begin{matrix}3 & 4\\2 & 5 \end{matrix}\right)\).
Solution:
The given matrix is A = \(\left(\begin{matrix}3 & 4\\2 & 5 \end{matrix}\right)\).
The formula to calculate the inverse of matrix for a matrix A = \(\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\) is A^{1} = \(\dfrac{1}{ad  bc}\left(\begin{matrix}d&b\\c&a\end{matrix}\right)\).
Using this formula we can calculate A^{1} as follows.
A^{1} = \(\dfrac{1}{(3)× 5  4 × 2}\left(\begin{matrix}5&4\\2&3\end{matrix}\right)\)
= \(\dfrac{1}{15  8}\left(\begin{matrix}5&4\\2&3\end{matrix}\right)\)
= \(\dfrac{1}{23}\left(\begin{matrix}5&4\\2&3\end{matrix}\right)\)
Answer: Therefore A^{1} = \(\dfrac{1}{23}\left(\begin{matrix}5&4\\2&3\end{matrix}\right)\)

Example 2: Find the inverse of the matrix A = \(\left(\begin{matrix}4 & 2 & 1\\5&0&3\\1&2 & 6\end{matrix}\right)\).
Solution:
The given matrix is A = \(\left(\begin{matrix}4 & 2 & 1\\5&0&3\\1&2 & 6\end{matrix}\right)\)
Step  1: Let us find the determinant of the given matrix using Row  1 of the above matrix.
A = \(4\left\begin{matrix}0&3\\2 & 6\end{matrix}\right (2)\left\begin{matrix}5&3\\1 & 6\end{matrix}\right+1\left\begin{matrix}5&0\\1& 2\end{matrix}\right\)
= 4(0 ×6  3×2) + 2(5×6  (1)×3) +1(5×2  0 ×(1))
= 4(0  6) +2(30 + 3) + 1(10  0)
= 24 + 66 + 10
= 52
Now, we will determine the adjoint of the matrix A by calculating the cofactors of each element and then taking the transpose of the cofactor matrix.
Adj A = \(\left(\begin{matrix}6 & 14 & 6\\33&25&7\\10&6 & 10\end{matrix}\right)\)
The inverse of matrix A is given by the formula A^{1} = \(\dfrac{1}{A}\).Adj A
A^{1} = \(\dfrac{1}{52}\).\(\left(\begin{matrix}6 & 14 & 6\\33&25&7\\10&6 & 10\end{matrix}\right)\)
= \(\left(\begin{matrix}3/26 & 7/26 & 3/26\\33/52&25/52&7/52\\5/26&3/26 & 5/26\end{matrix}\right)\)
Answer: A^{1} = \(\left(\begin{matrix}3/26 & 7/26 & 3/26\\33/52&25/52&7/52\\5/26&3/26 & 5/26\end{matrix}\right)\)
FAQs on Inverse of Matrix
What is the Inverse of Matrix?
The inverse of matrix is another matrix, which on multiplying with the given matrix gives the multiplicative identity. For a matrix A, its inverse is A^{1}, and A.A^{1 } = I. The general formula for the inverse of matrix is equal to the adjoint of a matrix divided by the determinant of a matrix.
A^{1} = \(\dfrac{1}{A}\).Adj A
The inverse of matrix exists only if the determinant of the matrix is a nonzero value.
How to find Inverse of Matrix?
The inverse of a square matrix is found in two simple steps. First, the determinant and the adjoint of the given square matrix are calculated. Further, the adjoint of the matrix is divided by the determinant to find the inverse of the square matrix. The inverse of the matrix A is equal to \(\dfrac{1}{A}\).Adj A.
How to find Inverse of a 2 x 2 Matrix?
The inverse of a 2 x 2 matrix is equal to the adjoint of the matrix divided by the determinant of the matrix. For a matrix A = \(\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\), its adjoint is equal to the interchange of the elements of the first diagonal and the sign change of the elements of the second diagonal. The formula for the inverse of the matrix is as follows.
A^{1} = \(\dfrac{1}{ad  bc}\left(\begin{matrix}d&b\\c&a\end{matrix}\right)\)
How to Use Inverse of Matrix?
The inverse of matrix is useful in solving equations by using the matrix inversion method. The matrix inversion method using the formula of X = A^{1}B, where X is the variable matrix, A is the coefficient matrix, and B is the constant matrix.
Can an Invertible Matrix be Called the Inverse of Matrix?
Yes, an invertible matrix can be called the inverse of matrix. The matrix whose determinant is not equal to zero is a nonsingular matrix. And for a nonsingular matrix, we can find the determinant and the inverse of matrix.
When Does the Inverse of Matrix Does not Exist?
The inverse of matrix exists only if its determinant value is a nonzero value. Because the adjoint of the matrix is divided by the determinant of the matrix, to obtain the inverse of a matrix. The matrix whose determinant is a nonzero value is called a nonsingular matrix.