# Xam Idea Class 12 Maths Chapter 4 Solutions: Determinants

Xam Idea Class 12 Maths Solutions Chapter 5 ‘Adjoint and Inverse of a Matrix’ is an extremely helpful book in getting your fundamentals of mathematics cleared for CBSE Class 12 board exams. The questions follow the new and revised CBSE exam pattern so that you are doing problems that are relevant to you from the board exams perspective. Our Xam Idea solutions for Adjoint and Inverse of a Matrix will also help you in preparing for many engineering entrance and competitive exams.

Xam Idea Class 12 Maths Solutions for** **Adjoint and Inverse of a Matrix have a total of 60 questions divided neatly into 8 exercises. You get your hands on some selected NCERT exemplar problems also in these sections along with some questions which have come in previous CBSE exams. The key ideas discussed here will help you understand what is the adjoint of a matrix and properties of the adjoint of a square matrix, what is an invertible matrix and how to find the inverse of a matrix, elementary row and column operations on a matrix, a number of solutions of a system of linear equations by examples, using the inverse of a matrix to solve a system of linear equations in two or three variables.

Doing well in mathematics needs rigorous practice and scoring good marks in CBSE exams needs you to practice the kind of questions that might be asked in the exams. Our team at Instasolv strives to always present the Xam Idea solutions in their simplest form so that you can go through them very easily. Our solutions, apart from teaching you concepts, will also provide you with strategies to solve more problems in less time so that you can do a better time-management during board exams.

## Important Topics for Xam Idea Class 12 Maths Solutions Chapter 5: Adjoint and Inverse of a Matrix

We had seen what the minor and cofactor of a matrix are in the previous chapter. That concept is used to further explain the adjoint of a matrix.

**Adjoint of a Matrix: **If A is a square matrix of order n and C_{ij} is the cofactor of element a_{ij} then transpose of C_{ij} is defined as adjoint of the matrix.

adj *A* = [*C _{ij}*]

^{T}

For a 3 X 3 matrix we can find its adjoint like below:

Adjoint of 2 X 2 square matrix can be obtained by interchanging the diagonal elements and changing the sign of non-diagonal elements.

**Some Important Properties of Adjoint of a Matrix: **For 2 matrices X and Y of the same order n

o Adj(XY) = adj(X) adj(Y)

o Adj X^{T} = (adj X)^{T}

o |adj X| = |X|^{n-1}

o Adj (adj X) = |X|^{n-2} X

**Invertible Matrix** – If for a square matrix A (of order n), there exists a matrix B of the same order such that AB = BA = I. Here matrix B is called the inverse of matrix A and represented by A^{-1}. Some properties of the inverse of a matrix are:

o For any invertible matrix, its inverse is unique

o For an invertible matrix A, (A^{-1})^{-1} = A

o Only non-singular square matrices can be invertible

o For 2 invertible matrices A and B, (AB)^{-1} = B^{-1} A^{-1}

o For an invertible matrix A, (A^{T})^{-1} = (A^{-1})^{T}

**Elementary Operations on a Matrix – **Following 3 elementary operations can be applied to the rows and columns of a matrix

o Multiply all elements of a row or a column by a non-zero scalar

o Interchanging any 2 rows or columns

o To elements of a row (or column) adding elements of another row (or column) and multiplying by a scalar.

**System of Linear Equations** – A system of linear equations can be represented in matrix form by using a coefficient matrix, variable matrix, and a constant matrix. We have to make sure we write the equation in a standard way so that the constant is on the right always.

Let the equations be: 2x + 3y = 8

5x – y = -2

We find the coefficient matrix by aligning coefficient of each equation in a row:

The variables here are x,y which can be written as a variable matrix:

The constants on the right can be represented as a constant matrix: :

This system can now be represented as:

Such representation makes calculations easier and it can be extended for n number of system linear equations.

### Discussion of Exercises of Xam Idea Class 12 Maths Solutions Chapter 5: Adjoint and Inverse of a Matrix

- Xam Idea Class 12 Maths Adjoint and the inverse of a Matrix’s first set of exercises has 3 questions that are taken from NCERT where the inverse of a matrix and system of linear equations has to be solved.
- The second set of exercises has 20 multiple-choice questions which would brush your knowledge of adjoint and inverse of a matrix. There are some questions on elementary row operations also.
- The third set of exercises is an assertion-reason type with one question where you have to evaluate the correctness of a statement.
- The fourth set is of very short answer type and has 7 questions. It has sums on the concept of an invertible matrix and adjoint of a matrix. It has some sums from previous year CBSE exams also.
- Its fifth set of exercises of Xam Idea solutions for Adjoint and Inverse of a Matrix has 4 questions that require short answers. All the questions are on the invertible matrix with some old CBSE questions.
- The sixth set is long answer type and has 7 questions. Your expertise on finding the inverse of a matrix by elementary operations and otherwise will be tested in this section. It also has a few word problems.
- The seventh set is again a long answer type and has 8 problems. The problems involve multiplying 2 matrices and finding their inverse, finding adjoint of matrices, and solving the system of equations using matrices. There are two-word problems given in this exercise.
- The last set is a self-assessment test section with 10 questions. They have a mix of question types from all parts of this chapter.

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- Xam idea Class 12 Maths Solutions gives you a unique learning experience with different parts providing you with fundamentals of the chapters, high order thinking questions, important questions from the previous year, and come from NCERT books.
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