Inverse of Matrix

The inverse of Matrix for a matrix A is denoted by A^-1. The inverse of a 2 × 2 matrix can be calculated using a simple formula. Further, to find the inverse of a matrix of order 3 or higher, we need to know about the determinant and adjoint of the matrix. The inverse of a matrix is another matrix, which by multiplying with the given matrix gives the identity matrix.

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 terms related to the inverse of matrix.

The inverse of matrix is a matrix, which on multiplication with the given matrix gives the multiplicative identity. For a square matrix A, its inverse is A^-1, and A · A^-1 = A^-1· A = I, where I is the identity matrix. The matrix whose determinant is non-zero and for which the inverse matrix can be calculated is called an invertible matrix. For example, the inverse of A = \(\left[\begin{array}{rr}
1 & -1 \\ \\
0 & 2
\end{array}\right]\) is \(\left[\begin{array}{cc}
1 & 1 / 2 \\ \\
0 & 1 / 2
\end{array}\right]\) as

• A · A^-1 = \(\left[\begin{array}{rr}
  1 & -1 \\ \\
  0 & 2
  \end{array}\right]\) \(\left[\begin{array}{cc}
  1 & 1 / 2 \\ \\
  0 & 1 / 2
  \end{array}\right]\) = \(\left[\begin{array}{cc}
  1 & 0 \\ \\
  0 & 1
  \end{array}\right]\) = I
• A^-1· A = \(\left[\begin{array}{cc}
  1 & 1 / 2 \\ \\
  0 & 1 / 2
  \end{array}\right]\) \(\left[\begin{array}{rr}
  1 & -1 \\ \\
  0 & 2
  \end{array}\right]\) = \(\left[\begin{array}{cc}
  1 & 0 \\ \\
  0 & 1
  \end{array}\right]\) = I

But how to find the inverse of a matrix? Let us see in the upcoming sections.

In the case of real numbers, the inverse of any real number a was the number a^-1, such that a times a^-1 equals 1. We knew that for a real number, the inverse of the number was the reciprocal of the number, as long as the number wasn't zero. The inverse of a square matrix A, denoted by A^-1, is the matrix so that the product of A and A^-1 is the identity matrix. The identity matrix that results will be the same size as matrix A.

The formula to find the inverse of a matrix is: A^-1 = 1/|A| · Adj A, where

• |A| is the determinant of A and
• Adj A is the adjoint of A

Since |A| is in the denominator of the above formula, the inverse of a matrix exists only if the determinant of the matrix is a non-zero value. i.e., |A| ≠ 0.

To find the inverse of a square matrix A, we use the following formula: A^-1 = adj(A) / |A|; |A| ≠ 0

where

• A is a square matrix.
• adj(A) is the adjoint matrix of A.
• |A| is the determinant of A.

☛Note: For a matrix to have its inverse exists:

• The given matrix should be a square matrix.
• The determinant of the matrix should not be equal to zero.

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 sum of the row and column elements in 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 the 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.

Non-Singular Matrix: A matrix whose determinant value is not equal to zero is referred to as a non-singular matrix. For a non-singular matrix |A| ≠ 0 and hence its inverse exists.

Adjoint of Matrix: The adjoint of a matrix is the transpose of the cofactor element matrix of the given matrix.

Rules 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.

The inverse of a 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 row or column transformations. Also, the inverse of a matrix can be calculated by applying the inverse of matrix formula through the use of the determinant and the adjoint of the matrix. For performing the inverse of the matrix through elementary column operations we use the matrix X and the second matrix B on the right-hand side of the equation.

• Elementary row or column operations
• Inverse of matrix formula (using the adjoint and determinant of matrix)

Let us check each of the methods described below.

Elementary Row Operations

To calculate the inverse of matrix A using elementary row transformations, we first take the augmented matrix [A | I], where I is the identity matrix whose order is the same as A. Then we apply the row operations to convert the left side A into I. Then the matrix gets converted into [I | A^-1]. For a more detailed process, click here.

Elementary Column Operations

We can apply the column operations as well just like how the process was explained for row operations to find the inverse of matrix.

Inverse of Matrix Formula

The inverse of matrix A can be computed using the inverse of matrix formula, A^-1 = (adj A)/(det A). i.e., by dividing the adjoint of a matrix by the determinant of the matrix. The inverse of a matrix can be calculated by following the given steps:

• Step 1: Calculate the minors of all elements of A.
• Step 2: Then compute the cofactors of all elements and write the cofactor matrix by replacing the elements of A by their corresponding cofactors.
• Step 3: Find the adjoint of A (written as adj A) by taking the transpose of the cofactor matrix of A.
• Step 4: Multiply adj A by the reciprocal of the determinant.

For a matrix A, its inverse A^-1 = \(\dfrac{1}{|A|}\) · Adj A

If A = \(\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\), then

• |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 = \(\dfrac{1}{|A|}.\begin{pmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\end{pmatrix}\)

In this section, we have learned the different methods to calculate the inverse of a matrix. Let us understand it better using a few examples for the different orders of matrices in the "examples" section below.

The inverse of 2 × 2 matrix is easier to calculate in comparison to matrices of higher order. We can calculate the inverse of 2 × 2 matrix using the general steps to calculate the inverse of a matrix. Let us find the inverse of the 2 × 2 matrix given below:
A = \(\begin{bmatrix} a & b \\ \\ c & d \end{bmatrix}\)
A^-1 = (1/|A|) × Adj A
= [1/(ad - bc)] × \(\begin{bmatrix} d & -b \\ \\ -c & a \end{bmatrix}\)

Therefore, in order to calculate the inverse of 2 × 2 matrix, we need to first swap the positions of terms a and d and put negative signs for terms b and c, and finally divide it by the determinant of the matrix.

We know that for every non-singular square matrix A, there exists an inverse matrix A^-1, such that A × A^-1 = I. Let us take any 3 × 3 square matrix given as,

A = \(\begin{bmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\)

The inverse of 3x3 matrix can be calculated using the inverse matrix formula, A^-1 = (1/|A|) × Adj A

We will first check if the given matrix is invertible, i.e., |A| ≠ 0. If the inverse of a matrix exists, we can find the adjoint of the given matrix and divide it by the determinant of the matrix.

A similar method can be followed to find the inverse of any n × n matrix. Let us see if similar steps can be used to calculate the inverse of m × n matrix where m ≠ n.

Inverse of 2 × 3 Matrix

We know that the first condition for the inverse of a matrix to exist is that the given matrix should be a square matrix. Also, the determinant of this square matrix should be non-zero. This means that the inverse of matrices of the order m × n will not exist where m ≠ n. Therefore, we cannot calculate the inverse of 2 × 3 matrix.

Inverse of 2 × 1 Matrix

Similar to the inverse of 2 × 3 matrix, the inverse of 2 × 1 matrix also does not exist because the given matrix is not a square matrix.

The determinant of the inverse of an invertible matrix is the inverse of the determinant of the original matrix. i.e., det(A^-1) = 1 / det(A). Let us check the proof of the above statement.

We know that, det(A • B) = det (A) × det(B)

Also, A × A^-1 = I

det(A × A^-1) = det(I)

or, det(A) × det(A^-1) = det(I)

Since, det(I) = 1

det(A) × det(A^-1) = 1

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

Hence, proved.

☛Related Articles:

• Inverse Matrix Calculator
• Matrix Multiplication Calculator
• Determinant Calculator

Important Points on Inverse of a 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^-1A^-1.
• The inverse of a square matrix A exists, only if its determinant is a non-zero value, |A| ≠ 0.
• The determinant of matrix inverse is equal to the reciprocal of the determinant of the original matrix.
• The determinant of the product of two matrices is equal to the product of the determinants of the two individual matrices. |AB| = |A|.|B|

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What is the Meaning of Inverse of Matrix?

The inverse of a matrix is another matrix, which multiplies with the given matrix and 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. i.e., A^-1 = 1/|A| · Adj A. The inverse of a matrix exists only if the determinant of the matrix is a non-zero value.

What is the Formula for An Inverse Matrix?

The inverse matrix formula is used to determine the inverse matrix for any given matrix. The inverse of a square matrix, A is A^-1 only when: A × A^-1 = A^-1 × A = I. The inverse matrix formula can be given as, A^-1 = adj(A)/|A|; |A| ≠ 0, where A is a square matrix.

How to Calculate Inverse of Matrix?

The inverse of a square matrix is calculated in 2 simple steps.

• Step 1: First, the determinant and the adjoint of the given square matrix are calculated.
• Step 2: The inverse of the matrix A is equal to \(\dfrac{1}{|A|}\).Adj A.

How to Find Inverse of a 2 × 2 Matrix?

The inverse of a 2 × 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^-1B, where X is the variable matrix, A is the coefficient matrix, and B is the constant matrix.

Can Inverse of Matrix be Calculated for an Invertible Matrix?

Yes, the inverse of matrix can be calculated for an invertible matrix. The matrix whose determinant is not equal to zero is a non-singular matrix. And for a non-singular matrix, we can find the determinant and the inverse of matrix.

When Does the Matrix Inverse not Exist in Some Cases?

The inverse of matrix exists only if its determinant value is a non-zero value and when the given matrix is a square matrix. 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 non-zero value is called a non-singular matrix. Matrix inverse is not defined for rectangular matrices.

Given a 2 × 2 Matrix, What is the Formula for Finding the Inverse of the Matrix?

For a given 2×2 matrix A = \(\left(\begin{matrix}a&b\\ \\c&d\end{matrix}\right)\) , inverse is given by A^-1 = \(\dfrac{1}{ad - bc}\left(\begin{matrix}d&-b\\\\-c&a\end{matrix}\right)\). Here A^-1 is the inverse of A.

What is the Inverse of Identity Matrix?

The inverse of an identity matrix is itself. This is because for any identity matrix of order I, we have I × I = I × I = I. For more information, click here.

How to Use Inverse Matrix Formula?

The inverse matrix formula can be used following the given steps:

• Step 1: Find the matrix of minors for the given matrix.
• Step 2: Then find the matrix of cofactors.
• Step 3: Find the adjoint by taking the transpose of the matrix of cofactors.
• Step 4: Divide it by the determinant.

What is 3 × 3 Inverse Matrix Formula?

The matrix inverse formula for a 3 × 3 matrix is, A^-1 = adj(A)/|A|; |A| ≠ 0 where A = square matrix, adj(A) = adjoint of square matrix, A^-1 = inverse matrix.

What is the Inverse of Diagonal Matrix?

The inverse of a diagonal matrix is again a diagonal matrix in which the elements of the principal diagonal of the matrix inverse are the reciprocals of the corresponding elements of the original matrix. To know how to prove this, click here.