Are these practice questions for Matrices based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 12 - Science Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Matrices free?

Yes. Every chapter includes free practice questions for Class 12 - Science students to get you started.

What types of questions are included?

The practice set for Matrices mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Matrices for full coverage.

How many practice questions are there for Matrices?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Matrices Class 12 Mathematics Practice Questions with Answers

easy

If matrix A has order

3 \times 4

and matrix B has order

4 \times 2

, analyze the orders of the product matrices AB and BA to determine which one is defined and what its order is.

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easy

Define a zero matrix (or null matrix). Can a zero matrix be a square matrix? Give an example of a

2 \times 3

zero matrix.

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easy

If

\begin{bmatrix} 2x+y \ 3 \end{bmatrix} = \begin{bmatrix} 7 \ x-y \end{bmatrix}

, calculate the values of

x

and

y

.

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easy

Find the transpose of the matrix

P = \begin{bmatrix} 1 & 0 & -2 \ 4 & 2 & 5 \ 3 & -1 & 7 \end{bmatrix}

.

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easy

If a matrix has 7 elements, list all its possible orders.

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easy

Construct a

2 \times 2

matrix

A = [a_{ij}]

whose elements are given by the formula

a_{ij} = (i - j)^3

.

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easy

Explain the concept of 'order of a matrix'. Given the matrix

A = \begin{bmatrix} \sqrt{3} & 1 & -5 & 17 \ 35 & -2 & \frac{5}{2} & 12 \end{bmatrix}

, identify its order and the total number of elements.

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easy

What is the condition for a matrix

A = [a_{ij}]_{m \times n}

to be a square matrix?

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easy

Justify why for any square matrix A, the matrix A - A' is always a skew-symmetric matrix.

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easy

Define a row matrix and give one example.

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easy

Create two non-zero 2 \times 2 matrices A and B such that their product AB is a zero matrix.

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easy

Summarize the rules for (a) addition of two matrices and (b) multiplication of a matrix by a scalar. Given

A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}

and

B = \begin{bmatrix} 0 & 5 \ 6 & 7 \end{bmatrix}

, find

A+B

and

3A

.

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medium

If

A = \begin{bmatrix} 2 & -1 \ 3 & 1 \end{bmatrix}

and

B = \begin{bmatrix} 1 & 4 \ 0 & -2 \end{bmatrix}

, calculate

3A - 2B

.

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medium

Define a symmetric matrix and a skew-symmetric matrix. What is the key difference regarding their diagonal elements?

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medium

Propose a proof for the uniqueness of the inverse of a square matrix. That is, if a square matrix A has an inverse, prove that it is unique.

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medium

For the matrices

A = \begin{bmatrix} 1 \ -4 \ 3 \end{bmatrix}

and

B = \begin{bmatrix} -1 & 2 & 1 \end{bmatrix}

, analyze and verify that

(AB)' = B'A'

.

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medium

A trust fund has ₹50,000 to invest in two different types of bonds. The first bond pays 6% interest per year, and the second bond pays 8% interest per year. Using matrix multiplication, formulate a method to determine how to divide the ₹50,000 among the two types of bonds to obtain an annual total interest of ₹3,600. Justify your formulation and solve for the amounts.

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medium

Evaluate the statement: 'If A is a symmetric matrix, then A^n is also a symmetric matrix for any positive integer n.' Justify your conclusion.

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medium

Let

A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}

and

B = \begin{bmatrix} 4 & 0 \ 1 & -2 \end{bmatrix}

. Calculate

AB

and

BA

and demonstrate that matrix multiplication is not commutative.

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medium

Solve for the matrix

X

if

2X + \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 8 \ 7 & 2 \end{bmatrix}

.

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medium

Identify the diagonal elements of the matrix

A = \begin{bmatrix} 3 & -1 & 0 \ \frac{3}{2} & 3\sqrt{2} & 1 \ 4 & 3 & -1 \end{bmatrix}

.

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medium

List the four main properties associated with the transpose of matrices.

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medium

A school wants to award its students for the values of Honesty, Regularity and Hard work. The number of students to be awarded in each category are 3, 2, and 1 respectively. The award money for each value is ₹1000, ₹1500, and ₹2000 per student. Calculate the total award money required using matrix multiplication.

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medium

Define the following types of matrices, each with a unique example of a

3 \times 3

matrix: (i) Diagonal Matrix (ii) Scalar Matrix (iii) Identity Matrix Explain the relationship between these three types of matrices.

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medium

Express the matrix

A = \begin{bmatrix} 3 & 5 \ 1 & -1 \end{bmatrix}

as the sum of a symmetric and a skew-symmetric matrix.

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medium

Solve the following matrix equations to find the matrices

X

and

Y

:

X + Y = \begin{bmatrix} 7 & 0 \ 2 & 5 \end{bmatrix}

X - Y = \begin{bmatrix} 3 & 0 \ 0 & 3 \end{bmatrix}

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medium

Formulate a 3 \times 3 matrix A which is neither symmetric nor skew-symmetric. Then, create the symmetric matrix P and skew-symmetric matrix Q such that A = P + Q and verify your result.

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medium

State the property for the existence of an additive identity in matrix addition.

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medium

Prove by the principle of mathematical induction that if A = \begin{bmatrix} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{bmatrix}, then A^n = \begin{bmatrix} \cos n\theta & \sin n\theta \ -\sin n\theta & \cos n\theta \end{bmatrix} for all n \in N.

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medium

If A and B are square matrices of the same order, justify why (A+B)^2 = A^2 + 2AB + B^2 is not always true. Propose the necessary condition for this equality to hold.

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medium

Describe the two conditions that must be met for two matrices,

A

and

B

, to be considered equal. Provide an example of two matrices that have the same order but are not equal.

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medium

If

A = \begin{bmatrix} 2 & 0 \ 1 & 5 \end{bmatrix}

and

I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}

, find the value of

k

such that

A^2 = 7A + kI

.

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medium

Find the real values of

a

and

b

if

A = \begin{bmatrix} a & b \ -b & a \end{bmatrix}

and

A^2 = \begin{bmatrix} 0 & 2 \ -2 & 0 \end{bmatrix}

.

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medium

If A and B are symmetric matrices of the same order, prove that AB - BA is a skew-symmetric matrix.

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medium

What is an invertible matrix? State the condition for a matrix

B

to be the inverse of a square matrix

A

.

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hard

Explain why matrix multiplication is not commutative in general. Use a simple example of

2 \times 2

matrices to illustrate your explanation.

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hard

Explain the process of multiplying two matrices,

A

and

B

. What is the essential condition on their orders for the product

AB

to be defined? If matrix

A

is of order

m \times n

and matrix

B

is of order

n \times p

, what will be the order of the product matrix

AB

?

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hard

Let A, B, and C be matrices of suitable orders such that their products are defined. Prove the distributive law A(B+C) = AB + AC.

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hard

Two families, A and B, have different monthly consumptions of rice, wheat, and sugar (in kg). Family A consumes 10 kg rice, 15 kg wheat, and 4 kg sugar. Family B consumes 12 kg rice, 18 kg wheat, and 5 kg sugar. The prices per kg are ₹40 for rice, ₹30 for wheat, and ₹45 for sugar. (i) Represent the consumption data as a

2 \times 3

matrix. (ii) Represent the price data as a

3 \times 1

matrix. (iii) Using matrix multiplication, calculate the monthly expenditure for each family.

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hard

If A is a square matrix satisfying the equation A^2 - 5A + 7I = O, prove that A is invertible and create an expression for A^{-1} in terms of A and I.

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hard

If A is an invertible square matrix, justify that (A')^{-1} = (A^{-1})'.

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hard

Show that the matrix B'AB is symmetric or skew-symmetric according as A is symmetric or skew-symmetric. Justify your proof for both cases.

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hard

If A is a square matrix such that A^2 = A (an idempotent matrix), prove that (I+A)^3 - 7A = I. Justify each step of the expansion.

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hard

If

A = \begin{bmatrix} 1 & 1 & 1 \ 1 & 2 & -3 \ 2 & -1 & 3 \end{bmatrix}

, calculate the value of the expression

A^3 - 6A^2 + 5A + 11I

, where I is the identity matrix of order 3, and analyze the result.

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hard

Create a 2 \times 2 matrix A that is its own inverse (A^{-1} = A), but is not the identity matrix I.

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MatricesClass 12 Mathematics Practice Questions