The Inverse of a Matrix

A matrix is a specific collection of elements organised in rows and columns. A matrix’s order is the number of rows x the number of columns. For instance, a 2*2 matrix means it contains two rows and two columns. We can only discover the matrix inverse for square matrices with equal numbers of rows and columns. The Inverse of a matrix is used to solve linear equations using the matrix inversion method. 

The inverse of a matrix X is represented by X-1. A simple formula may be used to determine the inverse of a 2*2 matrix. The inverse of a matrix is another matrix that yields the multiplicative identity when multiplied with the supplied matrix.

Let us now look at the formula, methods, and terminologies associated with the inverse of a matrix. 

What is the Inverse of the Matrix?

The Inverse of a matrix is another matrix that yields the multiplicative identity when multiplied by the supplied matrix. X-1 is the inverse of a matrix X, where X.X-1 = X-1. X = I, where I is the identity matrix. An invertible matrix has a non-zero determinant and for which the inverse matrix may be determined.

 For example, the inverse of X =

Is

Properties of Inverse Matrix

• The original matrix is equal to the inverse of the inverse matrix.

• If A and B are invertible matrices, then AB must be as well. As a result, (AB)-1 = B-1A-1

• If A is nonsingular, then (AT)-1 = (A-1)T

• A matrix’s product with its inverse, and vice versa, is always equal to the identity matrix.

Inverse Matrix Formula

In the case of real numbers, the inverse of every real number a was the number a-1, so that a multiplied by a-1 equaled 1. We understood that the inverse of a real number was the reciprocal of the number, as long as the number was not zero. The matrix is the inverse of a square matrix X, denoted by X-1, such that the product of X and X-1 equals the identity matrix. The resulting identity matrix will be the same size as matrix X.

A Matrix’s inverse:

X-1  =  1/|X|       

Because |X| is in the denominator of the expression, the inverse of the matrix exists only if the determinant of the matrix is non-zero. That is to say, |X|= 0.

The Inverse of Matrix Question

Some students may find it difficult to answer the Inverse of 3 by 3 Matrix. Hence, our question of the inverse of a matrix will be of a 3 by 3 matrix.

Steps to follow to find the inverse of a 3 by 3 matrix:

Calculate the determinant of a given matrix.

The first step is to compute the determinant of the 3 * 3 matrix, then discover its cofactors, minors, and adjoint, and then incorporate the findings into the inverse matrix formula shown below.

X−1=1/|X|Adj(X)

For example:

Check to see if the following matrix is invertible.

This may be demonstrated if the determinant is non-zero. There will be no inverse of the provided matrix if the determinant of the supplied matrix is zero.

det(X) = 1(0-24) – 2(0-20) + 3(0-5)

det(X) = -24 + 40 – 15

det (X) = 1

As the value determinant is 1, we can say that a given matrix has an inverse matrix.

• Determine the matrix’s transpose.
  
  To determine the transposition of the given 3 by 3 matrix.
  
  Hence, XT =

• Determine the determinant of the two-by-two matrix.

We will now calculate the determinant of each 2 X 2 minor matrices.

For 1st row elements:
 

Equals to -24
 

Equals to -18
 

For 2nd row elements:

Equals to -20

Equals to -15
 

Equals to 4

For 3rd row elements:

Equals to -5
 

Equals to -4
 

Equals to 1

The new matrix is:

• Create the cofactor matrix.
  
  Reverse the sign of the alternating terms to get the adjoint or adjugate matrix, as shown below:

As a result, we have the new matrix, X:

Adj (X) = the new matrix is :

Multiplied by : 

Adj (X) = 

• Finally, divide each adjugate matrix term by the determinant.

Determining the inverse of a 3 x 3 matrix:

Now, in the formula, we may swap the values of det (X) and adj (X):

X−1 = (1/det (X)) Adj (X)

The inverse of the matrix is:             X−1 = (1/1) =  

Conclusion 

By now you would have understood how to inverse a matrix and what are the steps involved in it. Typically you need to first find the determinant of a matrix. Then calculate the cofactor matrix. And, finally divide each adjugate matrix term by the determinant. The inverse of a Matrix is an important topic from the point of view of several competitive exams. Make sure to have a good grasp of this topic.