# Important Questions for CBSE Class 12 Maths Chapter 3 – Matrices 2023-24

## Important Questions For CBSE Class 12 Maths Chapter 3 – Matrices

Matrix is a powerful tool widely used in various branches of Maths. Chapter 3 of Class 12 Maths carries is important from an exam perspective. A comprehensive list of questions will help students focus on certain areas important from the exam point of view. However, students must prepare for all the subjects quickly before the exam. Therefore, subject matter experts have prepared a set of Important Questions Class 12 Maths Chapter 3 which will help them save their time and focus on more practice. These questions follow the CBSE pattern. Students can have a higher chance of scoring more marks by practising relevant questions provided by Extramarks.

You can also find CBSE Class 12 Maths Important Questions Chapter-by-Chapter Important Questions here:

 CBSE Class 12 Maths Important Questions Sr No Chapter No Chapter Name 1 Chapter 1 Relations and Functions 2 Chapter 2 Inverse Trigonometric Functions 3 Chapter 3 Matrices 4 Chapter 4 Determinants 5 Chapter 5 Continuity and Differentiability 6 Chapter 6 Application of Derivatives 7 Chapter 7 Integrals 8 Chapter 8 Application of Integrals 9 Chapter 9 Differential Equations 10 Chapter 10 Vector Algebra 11 Chapter 11 Three Dimensional Geometry 12 Chapter 12 Linear Programming 13 Chapter 13 Probability

## CBSE Class 12 Maths Chapter-3 Important Questions

Important Questions For CBSE Class 12 Maths Chapter 3 will help students learn the basics of matrices and solve more questions. The questions cover all the topics and help students revise the concepts crucial for calculations. Important Questions include the various types of matrices, their addition, subtraction and multiplication. This chapter is important, so students must understand the concepts well and solve more questions. Important Questions on Matrices will help students obtain more marks in the exam.

## Class 12 Maths Chapter 3 MatricesIntroduction

Matrix in maths refers to a set of numbers, either real or complex, arranged in a rectangular array. The numbers arranged here are enclosed ‘()’ or ‘||’. There are a few types of metrics a student of maths should know about.

### Types Of Matrices

1. Row Matrix: A Row Matrix is a matrix having only one row and any number of columns.
• Column Matrix: A matrix is said to be a Column Matrix if it has any number of rows but only one column.
• Horizontal Matrix: A matrix is called a Horizontal Matrix if the number of rows is lower than the number of columns.
• Vertical Matrix: In a Vertical Matrix the number of rows is higher than the number of columns.
• Scalar Matrix: A Scalar Matrix is a special type of Square Matrix where all the diagonal elements are equal and all the non-diagonal elements are zero.
• Square Matrix: A matrix which has an equal number of rows and columns is called a Square Matrix.
• Rectangular Matrix: In a Rectangular Matrix, the number of rows and columns are not equal.
• Diagonal Matrix: A Square Matrix is said to be a Diagonal Matrix if all of its non-diagonal elements are zero.
• Null/Zero Matrix: As the name suggests, the elements in a Null/Zero Matrix are all zero.
• Principal Diagonal Of A Matrix: In a Square Matrix, the straight path connecting the first diagonal element of the topmost row to the last element of the lowest row is called the Principal Diagonal of a Matrix.
• Identity/Unit Matrix: In a Square Matrix if all the diagonal elements are 1 and all the non-diagonal elements are zero, then it is called Identity/Unit Matrix.
• Singular Matrix: When the determinant of a Square Matrix is zero, then that matrix is called a Singular Matrix. If |A| = 0, it is a singular matrix. If |A| ≠ 0, it is a non-singular matrix.
• Equal Matrix: Two matrices are said to be equal if they have the same order and equal corresponding elements.

### Matrix Algebra

In this section, students will learn to perform certain operations on matrices, such as multiplication of matrices by scalar, addition and subtraction of matrices.

### Matrix Multiplication By Scalar

If A = [aij]mxn (where m x n is the size of the matrix) and k is a scalar quantity which is to be multiplied to A, the result of the multiplication is obtained when all the elements of A are multiplied by k.

Therefore, kA = [kaij]mxn .

Addition of two matrices is performed by adding the corresponding elements of the matrices provided that both the matrices have the same number of rows and same number of columns.

Therefore, if A = ajkmxn and B = bjkmxn

Then, A + B = [aij + bij]mxn .

### Matrix Subtraction

Suppose A and B are two matrices. A = ajkmxn and B = bjkmxn .

Then, the difference of these matrices

A – B = [aij – bij]mxn .

The only condition of Matrix Subtraction is that both matrices must have the same order.

### Some Chapter 3 Maths Class 12 Important Questions

1. Define Square Matrix.

Ans. A matrix with an equal number of rows and columns is called a Square Matrix, i.e. m = n.

1. What is meant by a Skew Matrix?

Ans. In a matrix, when the value of every diagonal element is zero, it is considered a Skew Matrix.

1. Write down the possible orders of a matrix having 28 elements.

Ans. The possible orders of a matrix having 28 elements are as follows:

• 1 X 28
• 2 X 14
• 4 X 7
• 7 X 4
• 14 X 2
• 28 X 1

Did You Know?

• Matrices are used in wireless communication to detect, extract and process the information embedded in the signals.
• They are also used in research activities to represent studies related to linear maps between the spaces of finite-dimensional vectors.

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### Conclusion

Matrix is a mathematical tool used for simplifying calculations. It has a long history of solving linear equations in compact and simple methods. In other words, a matrix can be defined as a set of numbers, symbols or expressions arranged in a tabular format. The horizontal sets are called rows and the vertical sets are known as columns.

The data denoted by numbers, expressions or symbols are called elements or entries of a matrix. The size of a matrix is determined by the number of its rows and columns. To add or subtract, the matrices concerned are to be of the same size. However, two matrices can be multiplied if the number of columns of the first matrix is equal to the number of rows of the second matrix.

There are several types of matrices. Matrices are used at higher levels of maths, machine learning, cryptography, computer graphics and many other branches of commerce and telecommunication industry.

To know important topics of Matrices in Class 12 Maths, students can access the Class 12 Maths Chapter 3 Important Questions from Extramarks.

Q1.

$\begin{array}{l}\text{If}\left[\begin{array}{cc}3\mathrm{a}–\mathrm{b}& \mathrm{a}+3\mathrm{b}\\ 2\mathrm{c}?\mathrm{d}& 4\mathrm{c}+\mathrm{d}\end{array}\right]=\left[\begin{array}{cc}5& 5\\ 10& 2\end{array}\right]\text{, then the value}\\ \text{of}\mathrm{a}\text{,}\mathrm{b},\mathrm{c}\text{and}\mathrm{d}\text{is}\end{array}$

Opt

1. a = 1, b = -2, c = 2 and d = -3.

2. a = 1, b = -2, c = 2 and d = -6.

3. a = -1, b = -8, c = 2 and d = -6.

4. a = -1, b = 8, c = 2 and d = -3.

Ans

$\begin{array}{l}\text{As the given matrices are equal, therefore, their corresponding}\\ \text{elements must be equal. Comparing the corresponding elements,}\\ \text{we get}\\ 3\mathrm{a}\mathrm{b}=5,\mathrm{a}+3\mathrm{b}=?5,\\ 2\mathrm{c}\mathrm{d}=10\text{?}\mathrm{and}\text{?}4\mathrm{c}+\mathrm{d}=2\\ \text{Solving these equations, we get}\\ \mathrm{a}=1,\mathrm{b}=?2\\ \mathrm{c}=2,\mathrm{d}=?6\end{array}$

Q2.

$\begin{array}{l}\mathrm{Let}\mathrm{A}=\left[\begin{array}{l}01\\ 00\end{array}\right],\mathrm{show}\mathrm{that}{\left(\mathrm{aI}+\mathrm{bA}\right)}^{\mathrm{n}}\mathrm{?}={\mathrm{a}}^{\mathrm{n}}\mathrm{I}+{\mathrm{na}}^{\mathrm{n}1}\mathrm{bA},\mathrm{where}\mathrm{I}\mathrm{is}\mathrm{the}\mathrm{identity}\\ \mathrm{matrix}\mathrm{of}\mathrm{order}2\mathrm{and}\mathrm{n}\mathrm{N}.\end{array}$

Opt.

$\begin{array}{l}\mathrm{We}\mathrm{shall}\mathrm{prove}\mathrm{the}\mathrm{result}\mathrm{by}\mathrm{using}\mathrm{the}\mathrm{principle}\mathrm{of}\mathrm{mathematical}\mathrm{induction}.\\ \mathrm{For}\mathrm{n}= 1,\mathrm{we}\mathrm{have}\\ \mathrm{P}\left(1\right):\left(\mathrm{al}+\mathrm{bA}\right)=\mathrm{al}+{\mathrm{ba}}^{}\mathrm{A}\\ =\mathrm{al}+\mathrm{bA}\\ \end{array}$

Ans.

$\begin{array}{l}\mathrm{We}\mathrm{shall}\mathrm{prove}\mathrm{the}\mathrm{result}\mathrm{by}\mathrm{using}\mathrm{the}\mathrm{principle}\mathrm{of}\mathrm{mathematical}\mathrm{induction}.\\ \mathrm{For}\mathrm{n}= 1,\mathrm{we}\mathrm{have}\\ \mathrm{P}\left(1\right):\mathrm{}\left(\mathrm{al}+\mathrm{bA}\right)\mathrm{}=\mathrm{al}+{\mathrm{ba}}^{}\mathrm{A}\\ =\mathrm{al}+\mathrm{bA}\\ \mathrm{Therefore},\mathrm{the}\mathrm{result}\mathrm{is}\mathrm{true}\mathrm{for}\mathrm{n}= 1.\\ \mathrm{Let}\mathrm{the}\mathrm{result}\mathrm{be}\mathrm{true}\mathrm{for}\mathrm{n}=\mathrm{k}\\ \mathrm{so},\\ \mathrm{P}{\left(\mathrm{al}+\mathrm{bA}\right)}^{\mathrm{k}}={\mathrm{a}}^{\mathrm{k}}\mathrm{l}+{\mathrm{ka}}^{\mathrm{k}1}\mathrm{bA}\mathrm{Now},\mathrm{we}\mathrm{will}\mathrm{prove}\mathrm{that}\mathrm{it}\mathrm{is}\mathrm{true}\mathrm{for}\mathrm{n}=\mathrm{k}+1\\ {\left(\mathrm{al}+\mathrm{bA}\right)}^{\mathrm{k}+1}={\left(\mathrm{al}+\mathrm{bA}\right)}^{\mathrm{k}}\left(\mathrm{al}+\mathrm{bA}\right)\\ =\mathrm{}\left({\mathrm{a}}^{\mathrm{k}}+{\mathrm{ka}}^{\mathrm{k}1}\mathrm{bA}\right)\mathrm{}\left(\mathrm{al}+\mathrm{bA}\right)\\ =\mathrm{}{\mathrm{a}}^{\mathrm{k}+1}\mathrm{l}+{\mathrm{ka}}^{\mathrm{k}}\mathrm{bA}+{\mathrm{a}}^{\mathrm{k}}\mathrm{bAl}+{\mathrm{ka}}^{\mathrm{k}1}{\mathrm{b}}^{2}{\mathrm{A}}^{2}\\ {\left(\mathrm{al}+\mathrm{bA}\right)}^{\mathrm{k}+1}={\mathrm{a}}^{\mathrm{k}+1}+\left(\mathrm{k}+1\right){\mathrm{a}}^{\mathrm{k}}\mathrm{bA}+{\mathrm{ka}}^{\mathrm{k}1}{\mathrm{b}}^{2}{\mathrm{A}}^{2}\dots ..\left(1\right)\\ \mathrm{Now},{\mathrm{A}}^{2}=\mathrm{}\left[\begin{array}{l}01\\ 00\end{array}\right]\mathrm{}\left[\begin{array}{l}01\\ 00\end{array}\right]\\ =\mathrm{}\left[\begin{array}{l}01\\ 00\end{array}\right]\mathrm{}=\mathrm{}0\mathrm{}\\ \mathrm{From}\mathrm{equation}\left(1\right),\mathrm{we}\mathrm{have}\\ {\left(\mathrm{al}+\mathrm{bA}\right)}^{\mathrm{k}+1}={\mathrm{a}}^{\mathrm{k}+1}+\left(\mathrm{k}+1\right){\mathrm{a}}^{\mathrm{k}}\mathrm{bA}+{\mathrm{ka}}^{\mathrm{k}1}{\mathrm{b}}^{2}\left(0\right)\\ =\mathrm{}{\mathrm{a}}^{\mathrm{k}+1}+\left(\mathrm{k}+1\right)\mathrm{}{\mathrm{a}}^{\mathrm{k}}\mathrm{bA}\\ \mathrm{Therefore},\mathrm{the}\mathrm{result}\mathrm{is}\mathrm{true}\mathrm{for}\mathrm{n}=\mathrm{k}+1.\\ \mathrm{Thus},\mathrm{by}\mathrm{the}\mathrm{Principle}\mathrm{of}\mathrm{Mathematical}\mathrm{Induction},\mathrm{we}\mathrm{have}\\ {\left(\mathrm{aI}+\mathrm{bA}\right)}^{\mathrm{n}}={\mathrm{a}}^{\mathrm{n}}\mathrm{I}+{\mathrm{na}}^{\mathrm{n}1}\mathrm{bA},\mathrm{for}\mathrm{all}\mathrm{n}ˆˆ\mathrm{N}.\end{array}$

Q3. A matrix has 24 elements. What are the possible orders it can have?

Opt. The possible orders of the matrix are:

1 24, 2 12, 3 8, 4 6, 6 4, 8 3, 12 2 and 24 1.

Ans. The possible orders of the matrix are:

1 24, 2 12, 3 8, 4 6, 6 4, 8 3, 12 2 and 24 1.

Q4.

$\begin{array}{l}\mathrm{Express}\mathrm{A}=\left[\begin{array}{l}331\\ 221\\ 452\end{array}\right]\mathrm{as}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{a}\mathrm{symmetric}\\ \mathrm{and}\mathrm{a}\mathrm{skew}–\mathrm{symmetric}\mathrm{matrix}.\end{array}$

Ans.

$\begin{array}{l}\mathrm{Here}\mathrm{A}‘ =\left[\begin{array}{l}324\\ 325\\ 112\end{array}\right]\\ \mathrm{Let}\mathrm{P}=\frac{1}{2}\left(\mathrm{A}+\mathrm{A}‘\right)\\ \frac{1}{2}\mathrm{}\left(\mathrm{A}+\mathrm{A}‘\right)\mathrm{}=\mathrm{}\frac{1}{2}\mathrm{}\left[\begin{array}{l}331\\ 221\\ 452\end{array}\right]+\left[\begin{array}{l}324\\ 325\\ 112\end{array}\right]\\ =\mathrm{}\frac{1}{2}\mathrm{}\left[\begin{array}{l}615\\ 14\mathrm{}4\\ 544\end{array}\right]\mathrm{}=\mathrm{}\left[\begin{array}{l}3\frac{1}{2}\frac{5}{2}\\ \frac{1}{2}22\\ \frac{5}{2}22\end{array}\right]\\ \mathrm{Now}\mathrm{P}‘ =\left[\begin{array}{l}3\frac{1}{2}\frac{5}{2}\\ \frac{1}{2}22\\ \frac{5}{2}22\end{array}\right]\mathrm{}=\mathrm{}\mathrm{P}\\ \mathrm{Thus}\mathrm{P}=\frac{1}{2}\mathrm{}\left[\mathrm{A}\mathrm{A}‘\right]\\ \frac{1}{2}\left[\mathrm{A}\mathrm{A}‘\right]\mathrm{}=\mathrm{}\frac{1}{2}\mathrm{}\left[\left[\begin{array}{l}331\\ 221\\ 452\end{array}\right]\left[\begin{array}{l}\mathrm{}324\\ \mathrm{}325\\ 112\end{array}\right]\right]\\ =\mathrm{}\frac{1}{2}\mathrm{}\left[\left[\begin{array}{l}053\\ 206\\ 360\end{array}\right]=\left[\begin{array}{l}\mathrm{}0\frac{5}{2}\frac{3}{2}\\ \frac{5}{2}30\\ \frac{3}{2}\mathrm{}30\end{array}\right]\right]\\ \mathrm{Then}\mathrm{Q}‘ =\left[\begin{array}{l}0\frac{5}{2}\frac{3}{2}\\ \frac{5}{2}03\\ \frac{3}{2}30\end{array}\right]\mathrm{}=\mathrm{}\mathrm{Q}\\ \mathrm{Thus}\mathrm{Q}=\frac{1}{2}\mathrm{}\left(\mathrm{A}\mathrm{A}‘\right)\mathrm{is}\mathrm{a}\mathrm{skew}\mathrm{symmetric}\mathrm{matrix}.\\ \mathrm{Now},\mathrm{P}+\mathrm{Q}=\left[\begin{array}{l}3\frac{1}{2}\frac{5}{2}\\ \frac{1}{2}22\\ \frac{5}{2}22\end{array}\right]+\left[\begin{array}{l}\mathrm{}0\frac{5}{2}\frac{3}{2}\\ \frac{5}{2}03\\ \frac{3}{2}\mathrm{}30\end{array}\right]\mathrm{}=\mathrm{}\left[\begin{array}{l}331\\ 221\\ 452\end{array}\right]\mathrm{}=\mathrm{}\mathrm{A}\end{array}$