Inverse Matrix: Overview, Questions, Preparation

A matrix is a definite set of objects organised in columns and rows. These objects are referred to as matrix components. The order of a matrix is entered by the number of rows compared to the number of columns.

The inverse of a matrix:

If A is a non-singular square matrix, n x n matrix A-1 exists, which is called the inverse matrix of A in such a way that the property is satisfied:

A.A^-1 = A^-1. A = I, where the matrix of identity is I.

An important condition: The square matrix can be non-singular to find the opposite matrix, the determinant value of which is not equal to zero.

When numbers are written in rows and columns, they form a matrix. The order of the matrix is represented as the number of rows by the number of columns. For example, 2 × 2, 2 × 3, 3 × 2, 3 × 3, 4 × 4 etc. 

However, the inverse of a matrix may be found when the specific matrix is a square matrix. A square matrix is a matrix whose number of columns and rows are equal, such as 2 × 2, 3 × 3, etc. To find the inverse of the matrix, the square matrix must be non-singular whose determinant value does not equal zero. 

Methods to Find the Inverse Matrix

There are three different methods to find the inverse of a matrix. 

1. Using determinants.
2. Using minors and cofactors of elements. 
3. Elementary Transformation.

There are mainly two ways to find an inverse of a matrix:

1. Determinant method:

To find the inverse of a matrix, by the determinant method, you should know how to calculate adjoint and the determinant of the matrix.

The inverse of the matrix is given by - 1/|determinant|. adj(A)...     where A is the matrix

2. Elementary transformation method: 

For finding the matrix by the elementary transformation method, we have to convert the matrix first into an identity matrix.  After that, write A = IA, where the identity matrix of the same order as A is I. Apply a row operation sequence before we have an identity matrix on the LHS and use the same elementary RHS operations to get I = BA. Matrix B on the RHS is matrix A's inverse. Write A = IA and apply column operations sequentially until I = AB is obtained, where B is the inverse matrix of A, to find the inverse of A using column operations.

Properties of Inverse of the Matrix:

The properties of inverse matrices are listed below.

 If A and B are nonsingular matrices, then the inverse matrix will have the following properties:

• (A^-1)^-1 = A
• (AB^)-1 = A^-1B^-1
• (ABC)^-1 = C^-1B^-1A^-1
• (A₁ A₂….A_n)^-1 = A_n^-1A_n-1^-1……A₂^-1A₁^-1
• (A^T)^-1 = (A^-1)^T
• (kA)^-1 = (1/k) A^-1

In the chapter ‘Matrix,’ you will learn about the inverse matrix and the different methods associated with it. The weightage of this chapter is 6-7 marks. The topic of the inverse of a matrix is covered in the chapter Matrices. The students get to learn about the fundamentals of matrix and matrix algebra. The chapter has a weightage of 13 marks.

1. Find the inverse of the matrix:

 A=  

Solution:   

Determinant = -1≠0 ..    means inverse is possible

Now we have to calculate the  adjoint ..    the adjoint matrix is equal to:

This is the adjoint matrix we get   

So the inverse of the matrix is given by:

A-1 = 1/|determinant|. Adj (A)

        = 

2. Find the inverse of the matrix:

Solution: The determinant of the matrix is = 1, so the inverse of the matrix is possible

Adjoint of the matrix = 

The inverse of the matrix = 

3. What is the inverse of 

Solution:

A⁻¹ = 

1/det x

Since determinant 

=16

Therefore, A⁻¹=1/16

=

4. Matrices A and B will be inverse of each other only if 

(A) AB = BA       (B) AB = BA = 0       (C) AB = 0, BA = I          (D) AB = BA = I 

Solution:  

Option (A)  AB = BA  is correct.

5. Find the inverse of the matrix if it exists.

By Applying, R1-> R1-R2, We get

6. Matrices A and B will be inverse of each other only if

A .AB=BA.

B. AB=BA=0.

C. AB=BA=I.

D. AB=0, BA=I.

Solution: 

The correct option is C.

We know that if A is a square matrix of order n, and if B is also the square matrix of order n, such that AB=BA=I, then B is the inverse of A and A is the inverse of B.
Therefore, A and B are the inverse of each other only if AB=BA= I.

Q: How can the opposite of the matrix be found?

Q: How can you find a 3x3 matrix inverse?

Q: What is a matrix’s right inverse?

Q: What is the inverse of the identity matrix?

Q: What is the cofactor matrix?

Q: What's going to happen when you reverse the matrix?

Q: Is the reciprocal of the matrix invertible?

Q: Is the inverse or reversal of the matrix unique?

Q: What can be the result when it is inverted?

Q: Has inverse of a matrix commutative nature?