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

Matrices arrange numbers or functions in rows and columns so that mathematical information can be written compactly.
In CBSE Class 12 Maths Chapter 3, students learn matrix operations, transpose, inverse and proof-based results.

Matrices is Chapter 3 of Class 12 Maths. A matrix is an ordered rectangular array of numbers or functions arranged in rows and columns. This chapter explains the order of a matrix, types of matrices, equality of matrices, matrix algebra, matrix multiplication, transpose of a matrix, symmetric matrix, skew symmetric matrix and invertible matrix.

Use these Important Questions Class 12 Maths Chapter 3 to revise definitions, properties, calculations and proof-based answers for the 2026–27 exams. Start with matrix order and types, then practise addition, subtraction, scalar multiplication, row-column multiplication, transpose properties, singular matrix ideas and inverse-based questions.

Key Takeaways

• Matrix: A matrix is an ordered rectangular array of numbers or functions.
• Order: A matrix with m rows and n columns has order m × n.
• Matrix multiplication: AB is defined only when columns of A equal rows of B.
• Inverse matrix: If AB = BA = I, then B is the inverse of A.

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Very Short Answer Questions for Class 12 Maths Chapter 3

Matrix definitions form the base of Chapter 3. These direct questions help revise rows, columns, order, entries and special matrices.

Q1. What is a matrix?

A matrix is an ordered rectangular array of numbers or functions.

The numbers or functions in a matrix are called elements or entries.

For example,

A = [ 2  3 ]

       [ 4  5 ]

This matrix has 2 rows and 2 columns. So, its order is 2 × 2.

Q2. What is the order of a matrix?

The order of a matrix is written as:

Number of rows × Number of columns

For example,

A = [ 1  2  3 ]

       [ 4  5  6 ]

This matrix has 2 rows and 3 columns. Therefore, the order of A is 2 × 3.

Q3. What is an element or entry of a matrix?

An element of a matrix is any number or function present in the matrix.

In A = [aij], the element aij lies in the ith row and jth column.

For example, a23 means the element in the second row and third column.

Q4. What is a row matrix?

A row matrix is a matrix with only one row.

For example,

A = [ 2  5  7 ]

This is a row matrix of order 1 × 3.

Q5. What is a column matrix?

A column matrix is a matrix with only one column.

For example,

B = [ 3 ]

       [ 6 ]

       [ 9 ]

This is a column matrix of order 3 × 1.

Q6. What is a square matrix?

A square matrix is a matrix in which the number of rows is equal to the number of columns.

For example,

A = [ 1  2 ]

       [ 3  4 ]

This is a square matrix of order 2.

Q7. What is an identity matrix?

An identity matrix is a square matrix in which all diagonal elements are 1 and all non-diagonal elements are zero.

It is denoted by I.

For example,

I = [ 1  0 ]

      [ 0  1 ]

This is an identity matrix of order 2.

Q8. What is a zero matrix?

A zero matrix is a matrix in which every element is zero.

For example,

O = [ 0  0 ]

       [ 0  0 ]

This is a zero matrix of order 2 × 2.

Objective Questions from Class 12 Maths Chapter 3 Matrices

Objective questions from Matrices often check exact definitions, order and basic properties. Read the order of the matrix before choosing an option.

Q9. Choose the correct A matrix with m rows and n columns has order ______.

1. a) n × m
   b) m × n
   c) m + n
   d) mn × mn
2. b) m × n.

The order of a matrix is always written as rows × columns.

Q10. Choose the correct The number of elements in a 3 × 4 matrix is ______.

1. a) 7
   b) 9
   c) 12
   d) 34
2. c) 12.

A matrix of order m × n has mn elements.

So,

3 × 4 = 12

Q11. Fill in the blank: If A is a skew symmetric matrix, then all diagonal elements of A are ______.

zero.

For a skew symmetric matrix:

A' = -A

For diagonal elements:

aii = -aii

2aii = 0

aii = 0

Hence, all diagonal elements of a skew symmetric matrix are zero.

Q12. Fill in the blank: Matrix multiplication AB is defined when the number of columns of A is equal to the number of ______ of B.

rows.

If A has order m × n and B has order n × p, then AB is defined.

The order of AB is m × p.

Q13. True or False: Matrix multiplication is always commutative.

False.

Matrix multiplication is not generally commutative.

In most cases:

AB ≠ BA

Sometimes AB may be defined, while BA may not be defined.

Q14. True or False: Every identity matrix is a scalar matrix.

True.

An identity matrix is a scalar matrix in which all diagonal elements are equal to 1 and all non-diagonal elements are zero.

Short Answer Questions from Class 12 Maths Chapter 3 Important Questions

Short answer questions from Matrices usually involve construction, equality and basic operations. Show the entries and then write the final matrix.

Q15. If a matrix has 24 elements, what are the possible orders it can have?

If a matrix has 24 elements, then:

mn = 24

The possible orders are formed using factor pairs of 24.

1 × 24, 2 × 12, 3 × 8, 4 × 6, 6 × 4, 8 × 3, 12 × 2, 24 × 1

Hence, a matrix with 24 elements can have 8 possible orders.

Q16. Construct a 2 × 2 matrix A = [aij], where aij = (i + j) / 2.

A 2 × 2 matrix has four entries.

A = [ a11  a12 ]

       [ a21  a22 ]

Now calculate each entry using aij = (i + j) / 2.

a11 = (1 + 1) / 2 = 1

a12 = (1 + 2) / 2 = 3/2

a21 = (2 + 1) / 2 = 3/2

a22 = (2 + 2) / 2 = 2

Therefore,

A = [ 1    3/2 ]

       [ 3/2  2   ]

Q17. Construct a 3 × 2 matrix whose elements are given by aij = 1/2 |2i - 3j|.

A 3 × 2 matrix has 3 rows and 2 columns.

A = [ a11  a12 ]

       [ a21  a22 ]

       [ a31  a32 ]

Using aij = 1/2 |2i - 3j|:

a11 = 1/2 |2(1) - 3(1)| = 1/2

a12 = 1/2 |2(1) - 3(2)| = 2

a21 = 1/2 |2(2) - 3(1)| = 1/2

a22 = 1/2 |2(2) - 3(2)| = 1

a31 = 1/2 |2(3) - 3(1)| = 3/2

a32 = 1/2 |2(3) - 3(2)| = 0

Therefore,

A = [ 1/2  2 ]

       [ 1/2  1 ]

       [ 3/2  0 ]

Q18. Find x and y, if

[ x + 2  5     ] = [ 7  5 ]

[ 3      y - 1 ]   [ 3  4 ]

Equal matrices have equal corresponding elements.

Comparing the first row, first column:

x + 2 = 7

x = 5

Comparing the second row, second column:

y - 1 = 4

y = 5

Therefore,

x = 5, y = 5

Q19. Add the matrices A and B, where

A = [ 2  4 ]

       [ 3  5 ]

B = [ 1  6 ]

       [ 7  2 ]

Matrices of the same order are added by adding corresponding elements.

A + B = [ 2 + 1  4 + 6 ]

                [ 3 + 7  5 + 2 ]

Therefore,

A + B = [ 3   10 ]

                [ 10  7  ]

Q20. Find 2A - B, where

A = [ 1  2  3 ]

       [ 2  3  1 ]

B = [ 3  -1   3 ]

       [ 1   0  -2 ]

First find 2A.

2A = 2 [ 1  2  3 ]

            [ 2  3  1 ]

So,

2A = [ 2  4  6 ]

        [ 4  6  2 ]

Now subtract B from 2A.

2A - B = [ 2  4  6 ] - [ 3  -1   3 ]

                 [ 4  6  2 ]   [ 1   0  -2 ]

2A - B = [ 2 - 3  4 - (-1)  6 - 3   ]

                 [ 4 - 1  6 - 0     2 - (-2)]

Therefore,

2A - B = [ -1  5  3 ]

                 [ 3   6  4 ]

Matrix Multiplication Questions for Class 12 Maths Chapter 3

Matrix multiplication uses rows of the first matrix and columns of the second matrix. These questions apply the row-column rule.

Q21. State the condition for multiplication of two matrices.

The product AB is defined only when the number of columns of A is equal to the number of rows of B.

If A has order m × n and B has order n × p, then AB is defined.

The product AB will have order m × p.

Q22. Find AB, where

A = [ 1  2 ]

       [ 3  4 ]

B = [ 5  6 ]

       [ 7  8 ]

Use row-column multiplication.

AB = [ 1  2 ] [ 5  6 ]

          [ 3  4 ] [ 7  8 ]

First row, first column:

1(5) + 2(7) = 5 + 14 = 19

First row, second column:

1(6) + 2(8) = 6 + 16 = 22

Second row, first column:

3(5) + 4(7) = 15 + 28 = 43

Second row, second column:

3(6) + 4(8) = 18 + 32 = 50

So,

AB = [ 19  22 ]

          [ 43  50 ]

Q23. Show with an example that AB may be defined but BA may not be defined.

Let A be of order 2 × 3 and B be of order 3 × 4.

Then AB is defined because the number of columns of A is 3 and the number of rows of B is also 3.

For AB:

(2 × 3)(3 × 4) = 2 × 4

For BA:

(3 × 4)(2 × 3)

Here, 4 is not equal to 2. Hence, BA is not defined.

Q24. Show with an example that matrix multiplication is not commutative.

Let:

A = [ 1  0  ]

       [ 0  -1 ]

B = [ 0  1 ]

       [ 1  0 ]

Now,

AB = [ 1  0  ] [ 0  1 ]

          [ 0  -1 ] [ 1  0 ]

So,

AB = [ 0   1 ]

          [ -1  0 ]

And,

BA = [ 0  1 ] [ 1  0  ]

        [ 1  0 ] [ 0  -1 ]

So,

BA = [ 0  -1 ]

          [ 1   0 ]

Since the two products are different:

AB ≠ BA

Hence, matrix multiplication is not generally commutative.

Q25. Find the transpose of the matrix

A = [ 2  3  5 ]

       [ 1  4  6 ]

The transpose is obtained by changing rows into columns.

A' = [ 2  1 ]

        [ 3  4 ]

        [ 5  6 ]

Here, A is of order 2 × 3, while A' is of order 3 × 2.

Q26. Verify that (A')' = A, where

A = [ 1  2 ]

       [ 3  4 ]

First find A'.

A' = [ 1  3 ]

        [ 2  4 ]

Now transpose A' again.

(A')' = [ 1  2 ]

            [ 3  4 ]

This is the same as A. Therefore:

(A')' = A

Hence verified.

Q27. Verify that (A + B)' = A' + B', where

A = [ 1  2 ]

       [ 3  4 ]

B = [ 5  6 ]

       [ 7  8 ]

First find A + B.

A + B = [ 1 + 5  2 + 6 ]

               [ 3 + 7  4 + 8 ]

So,

A + B = [ 6   8  ]

                [ 10  12 ]

Now,

(A + B)' = [ 6  10 ]

                   [ 8  12 ]

Next,

A' = [ 1  3 ]

        [ 2  4 ]

B' = [ 5  7 ]

        [ 6  8 ]

So,

A' + B' = [ 1 + 5  3 + 7 ]

                [ 2 + 6  4 + 8 ]

Therefore,

A' + B' = [ 6  10 ]

                  [ 8  12 ]

Hence,

(A + B)' = A' + B'

Symmetric and Skew Symmetric Matrix Questions for Class 12 Maths Chapter 3

Symmetric and skew symmetric matrices are square matrices connected with transpose. These results are often asked as proof-based questions.

Q28. Show that A + A' is symmetric for any square matrix A.

Let:

B = A + A'

To prove that B is symmetric, we need to prove:

B' = B

Now take transpose of both sides.

B' = (A + A')'

Using (A + B)' = A' + B':

B' = A' + (A')'

Since (A')' = A:

B' = A' + A

By commutative property of matrix addition:

B' = A + A'

But B = A + A'. Therefore:

B' = B

Hence, A + A' is symmetric.

Q29. Express the square matrix A as the sum of a symmetric and a skew symmetric matrix, where

A = [ 2  4 ]

       [ 1  3 ]

Any square matrix A can be written as:

A = 1/2(A + A') + 1/2(A - A')

First find A'.

A' = [ 2  1 ]

        [ 4  3 ]

Now,

A + A' = [ 2  4 ] + [ 2  1 ]

               [ 1  3 ]   [ 4  3 ]

So,

A + A' = [ 4  5 ]

                 [ 5  6 ]

Therefore,

1/2(A + A') = [ 2    5/2 ]

                      [ 5/2  3   ]

This is the symmetric part.

Now,

A - A' = [ 2  4 ] - [ 2  1 ]

              [ 1  3 ]   [ 4  3 ]

So,

A - A' = [ 0   3 ]

              [ -3  0 ]

Therefore,

1/2(A - A') = [ 0     3/2 ]

                         [ -3/2  0   ]

This is the skew symmetric part.

Hence,

A = [ 2    5/2 ] + [ 0     3/2 ]

       [ 5/2  3   ]   [ -3/2  0   ]

Q30. Prove that the inverse of a square matrix, if it exists, is unique.

Let A be a square matrix. Suppose B and C are both inverses of A.

Since B is an inverse of A:

AB = BA = I

Since C is also an inverse of A:

AC = CA = I

Now start with B:

B = BI

Since AC = I, replace I with AC.

B = B(AC)

Using associativity of matrix multiplication:

B = (BA)C

Since BA = I:

B = IC

So:

B = C

Therefore, the inverse of a square matrix, if it exists, is unique.

Important Questions Class 12 Maths Chapter-Wise