Inverse Matrix

Matrices are an important topic in terms of class 11 mathematics. In that, most weightage is given to inverse matrix problems. Let us first define the inverse of a matrix.

According to the inverse of a matrix definition, a square matrix A of order n is said to be invertible if there exists another square matrix B of order n such that AB = BA = I.

where I is the identity of order n*n. Identity matrix of order 2 is denoted by

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Rank of The Matrix -

The rank of the matrix is the extreme number of linearly self-determining column vectors within the matrix.

How to find the inverse of a matrix/ how to determine the inverse of a matrix?

The inverse matrix can be found only with the square matrix. The square matrix has to be non-singular, i.e, its determinant has to be non-zero. A common question arises, how to find the inverse of a square matrix? By inverse matrix definition in math, we can only find inverses in square matrices.

Given a square matrix A

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Its determinant value is given by [(a*d)-(c*d)].

Some important results -

  • The inverse of a square matrix, if exists, is unique

  • AA-1= I= A-1a

  • If A and B are invertible then (AB)-1=  B-1A-1

  • Every orthogonal matrix is invertible

  • If A is symmetric then its inverse is also symmetric.

Broadly there are two ways to find the inverse of a matrix:

  1. Using Determinants -

This matrix inversion method is suitable to find the inverse of the 2 by 2 matrix.

A-1= (1/|A|)*adj(A)

where adj(A) refers to the adjoint matrix A, |A| refers to the determinant of a matrix A.

adjoint of a matrix is found by taking the transpose of the cofactor matrix.

Cij = (-1)ij det (Mij),  Cij is the cofactor matrix

  1. By Elementary Transformation -

This method is suitable to find the inverse of the n*n matrix.

In this method first, write A=IA if you are considering row operations, and A=AI if you are considering column operation.

Let us take 3 matrices X, A, and B such that X = AB. To find out the required identity matrix we find out using elementary operations and reduce to an identity matrix

If A-1 exists then to find A-1 using elementary row operations is as follows:

1.      Write A = IA, where I is the identity matrix as order as A.

2.      Validate the sum by performing the necessary row operations on LHS to get I in LHS. Make sure to perform the same operations on RHS so that you get I=BA.

If A-1 exists then to find A-1 using elementary column operations is as follows:

1.      Write A = AI, where I is the identity matrix as order as A.

2.      Validate the sum by performing the necessary column operations on LHS to get I in LHS. Make sure to perform the same operations on RHS so that you get I=AB.

The identity matrix of n*n is represented in the figure below

The identity matrix In is a n x n square matrix with the main diagonal of 1’s and all other elements are O’s.

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If A is a m x n matrix, then 

ImA = A and AIn = A

If A is a n x n matrix, then 

AIn = InA = A

Let us see how to do inverse matrix with examples of inverse matrix problems to understand the concept clearly

An inverse matrix example using the 1st method is shown below -

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An example of finding an inverse matrix with elementary column operations is given below

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An example of finding an inverse matrix with elementary row operations given below -

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Properties of Inverse of a Square Matrix

  • Square matrix A is invertible if and only if |A|≠ 0

  • (A-1)-1=A

  • (A’)-1 = (A-1)’

  • (AB)-1 = B-1A-1 In general ( A1A1A1 … An)-1 =  An-1An – 1-1   …  A3-1A2-1A1-1

  • Also, if a non singular square matrix A is symmetric, then A-1 is also symmetric.

  • |A-1| = |A|-1

  • AA-1 = A-1A = I

  • (Ak)-1 = (A-1) Akk ∈ N

  • If  abc ≠ 0, then A-1 = \[\begin{bmatrix} 1/a & 0 & 0\\ 0 & 1/b & 0 \\ 0 & 0 & 1/c\end {bmatrix}\]

Here’s Something For More Clarity –

How can we find the inverse of a 3 x 3 matrix?

In order to figure out the inverse of the 3 x 3 matrix, first of all, we need to determine the determinant of the matrix. If the determinant will be zero, the matrix will not be having any inverse. Then move the matrix by re-writing the first row as the first column, the middle row as the main column, and the third row as the third column.

Similarly,

What's The Inverse of a 4 x 4 Matrix?

The inverse of the 'n' x 'n' matrix 'A' is the 'n' x 'n' matrix 'B.' Like 'AB' = 'BA' = 'I.' And if we get the inverse of the 4 x 4 matrix 'A' to be 'B,' then we'll only have to multiply 'AB' and 'BA' to test our work.

How Are We Going To Measure The Inverse?

We can calculate the inverse of the matrix in the following steps-

1st Step - Calculate a minor matrix.

2nd Step - Then convert it to a cofactor matrix.

3rd Step -  Adjugate, then.

4th Step  - Finally, multiply with 1 / Determinant.

FAQ (Frequently Asked Questions)

1.  How important is this topic?

Answer  - This topic plays a major role in this chapter. The inverse of a matrix is a definite 4 mark question which you can attempt easily once you have mastered it. Also one has to be very careful while using the elementary transformation. Keeping this in mind double-check whether you are applying row or column operation. Read the question twice before applying the solution.

2. What are the important questions in the matrix chapter?

Answer  -

  • Elementary transformations

  • The inverse of a matrix

  • Algebra of matrices