# RS Aggarwal Class 12 Chapter 7 Solutions (Adjoint And Inverse of A Matrix)

RS Aggarwal Solutions for Class 12 Chapter 7 Adjoint and Inverse of a Matrix have been curated to bring you at ease with the important questions. This is a very important chapter from the RS Aggarwal Solutions for Class 12 Board exam point of view. You will learn to find the adjoint of a given matrix and also the inverse of any matrix using some equations and elementary operations, also a variety of properties of the inverse of Matrices are discussed in exercise solutions.

There are 37 questions and one exercise in this chapter which will give you an opportunity for rigorous practice of various topics. Going through these solutions will help you learn to optimise your time and avoid silly mistakes. A detailed answer with all the operations mentioned clearly is given for each exercise question to avoid general confusion that arises often while finding the Adjoint and Inverse of a Matrix. These questions will build a strong foundation of Matrices which is important not only for higher education but also for your CBSE Board Exams.

The expert team of Maths at Instasolv have curated easy answers for the exercises to help you with a quick revision. These solutions by Instasolv are indispensable if you are appearing in the 12th board examination.

## Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 7

**Adjoint And Inverse Of A Matrix**

**Adjoint:**

- If A = [a
_{ij}] is a square matrix of order n and C_{ij}denotes the cofactor of a_{ij}in A, then the transpose of the matrix of cofactors of elements of A is called the adjoint of A and is represented by adj A.

i.e., adj A = [a_{ij}]^{T}.

- The adjoint of a square matrix of order can be obtained by interchanging the diagonal elements and changing the signs of off-diagonal elements.
- For a square matrix A of order n, A(adj A) = | A | I
_{n}= (adj A) A - Following are some properties of the adjoint of a square matrix:
- For two square matrices A and B of the order n
- adj (AB) = (adj B) (adj A)
- adj A
^{T}= (adj A)^{T} - Adj (adj A) = | A |
^{-1}A - | adj A | = | A |
^{n}^{-1}

- For two square matrices A and B of the order n

**Inverse:**

- A square matrix A of order n is inevitable if there exists a square matrix B of the same order such that AB = I
_{n}= BA.

In such a case, we say that matrix B will be the inverse of matrix A and we write A^{-1} = B.

Properties of the inverse of a matrix are as follows:

- Every invertible matrix possesses a unique inverse.
- For an invertible matrix A, (A
^{-1})^{-1}= A - Only non-singular square matrices are invertible.
- If A is a non-singular matrix, then
- For two invertible matrices A and B of the same order, (AB)
^{-1}= B^{-1}A^{-1} - For an invertible matrix A, (A
^{T})^{-1}= (A^{-1})^{T} - Invertible symmetric matrix, when inverted gives a symmetric matrix.
- For a non-singular matrix

- The following are three operations applied on the rows/columns of a matrix:
- Interchange of any two rows/columns
- Multiplying all of the elements of a row/column of a matrix by a non zero scalar.
- Adding to the elements of a row/column, the corresponding elements of any other row/column multiplied by any scalar.

- A matrix obtained from an identity matrix by a single elementary operation is called an elementary matrix.
- Every Row/column operation on an m x n matrix except identity matrix can be obtained by pre multiplication/post multiplication with the corresponding elementary matrix obtained from the identity matrix I
_{m}/I_{n}by submitting it to the same elementary row/column operation. - To find the inverse of a non-singular square matrix A by elementary operations method, we use the following equation

A = IA

Now, a progression of elementary row operations are successively performed on A on the LHS and the pre-factor I on RHS, till we obtain

I = BA

The matrix B, so obtained, is the desired inverse of matrix A.

**Highlights of the Chapter 7 Adjoint and Inverse of a**

- The transpose of the matrix of cofactors of all the elements of a matrix A is termed as the adjoint of A and is represented as adj A.
- If AB = In = BA, in that case, the inverse of A is given as A
^{-1}= B

- To find the inverse, we can use either of the two operations: A = IA; I = BA.

### Discussion of Exercise Questions of RS Aggarwal Solutions for Class 12 Chapter 7 Adjoint and Inverse of a Matrix

- There is one exercise 7A with robust coverage of all the topics of this chapter. You will be required to find the adjoint of the matrices and their inverse.
- Some questions will also require you to apply the properties of the inverse of a matrix.
- There is short answer type of objective questions as well as long answers type questions which are subjective in nature.
- The questions are in increasing order of difficulty level to match your compatibility with the topics.
- You will get an idea of the pattern of questions asked in your exam.

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- The solutions provided by the team at Instasolv are in the simplest language possible to suit your objective of scoring good marks.
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- These solutions of the RS Aggarwal Class 12 Maths Solutions will optimise your time in the revision before your exam.
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- The solutions are strictly based on the Class 12 CBSE Board guidelines and therefore, are very reliable.