# Maths MCQs for Class 12 with Answers Chapter 3 Matrices

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## Matrices Class 12 Maths MCQs Pdf

1. If a matrix has 6 elements, then number of possible orders of the matrix can be
(a) 2
(b) 4
(c) 3
(d) 6

Explaination: (b), as 6 → 1 × 6, 2 × 3, 3 × 2, 6 × 1.

2. If A = [aij] is a 2 × 3 matrix, such that aij = $$\frac{(-i+2 j)^{2}}{5}$$.then a23 is Explaination: (d), as a23 = $$\frac{(-2+6)^{2}}{5}=\frac{16}{5}$$

3. If A = diag(3, -1), then matrix A is Explaination: (c), as diag (3, -1) is a diagonal matrix. Its order is 2 × 2 with diagonal elements 3 and (-1).

4. Total number of possible matrices of order 2 × 3 with each entry 1 or 0 is
(a) 6
(b) 36
(c) 32
(d) 64

Explaination: (d), as total elements are 6 and each entry can be done in 2 ways. Hence, total possibilities = 26 = 64.

5. If A is a square matrix such that A²=A, then (I + A)² – 3A is
(a) I
(b) 2A
(c) 3I
(d) A

Explaination: (a), as (I + A)² -3A = I² + IA + AI + A² – 3A = I + A + A + A – 3A=I

6. If matrices A and B are inverse of each other then
(a) AB = BA
(b) AB = BA = I
(c) AB = BA = 0
(d) AB = 0, BA = I

Explaination: (b), by definition.

7. Explaination: 8. The diagonal elements of a skew symmetric matrix are
(a) all zeroes
(b) are all equal to some scalar k(≠ 0)
(c) can be any number
(d) none of these

Explaination:
(a), as in skew symmetric matrix, aij = -aji
⇒ aii = – aii
⇒ 2aii = 0
⇒ aii = 0, i.e. diagonal elements are zeroes.

9. If A = $$\left[\begin{array}{ll}{5} & {x} \\ {y} & {0}\end{array}\right]$$ and A = A’ then
(a) x = 0, y = 5
(b) x = y
(c) x + y = 5
(d) x – y = 5

Explaination: 10. If a matrix A is both symmetric and skew symmetric then matrix A is
(a) a scalar matrix
(b) a diagonal matrix
(c) a zero matrix of order n × n
(d) a rectangular matrix.

Explaination: (c), as it satisfies aij = aji, = -aji and aii = 0

11. Explaination: False, as their orders are different.

12. If a matrix has 5 elements, write all possible orders it can have. [AI2011]

Explaination: Possible orders are 1 × 5, 5 × 1.

13. If A is a 3 × 3 matrix, whose elements are given by aiij = $$\frac{1}{3}$$|-3i + j|, then write the value a23. [Foreign 2013]

Explaination: Given aij = $$\frac{1}{3}$$|-3i + j|
∴ a23 = $$\frac{1}{3}$$|-6 + 3| = 1.

14. Explaination: 15. Explaination: x + y + z = 9, x + z = 5, y + z = 1, on solving we get
x = 2, y = 4, z = 3

16. Explaination:
cos α = 1, sin α = 0
⇒ α = 0°

17. Explaination: 3a22 – 4a33 = 3 × 5 – 4 × 4 = -1

18. Explaination: These matrices are not equal as their orders are not same.

19. Explaination: 20. If matrix X is such that then order of matrix X is ______ .

Explaination: 2 × 2, as (2 × 2) × (2 × 3) = 2 × 3

21. If A and B are matrices of order 3 × m and 3 × n respectively such that m = n, then order of 2A + 7B is _____ .

Explaination: 3 × m or 3 × n, as order of sum of two matrices is same as order of given matrices.

22. Matrix multiplication is commutative, state true or false.

Explaination: False, as AB ≠ BA in general.

23. The negative of matrix is obtained by multiplying the matrix by _____ .

Explaination: -1, as (-1)A = -A.

24. Explaination: a22 + b21 = 4 – 3 = 1

25. then write the value of k.

Explaination: 26. Explaination: 27. Find x and y, if Explaination: 28. find the value of x [Delhi 2012]

Explaination: 29. Explaination: 30. Find a matrix X such that 2A + B + X = Explaination:  31. If A is a square matrix such that A² = A, then write the value of (I + A)² – 3A. [Foreign 2012]

Explaination: (I + A)² – 3A = I² + IA+ AI + A² -3A = I + A+ A+ A – 3A = I

32. Explaination: 33. If matrix A = $$\left[\begin{array}{cc}{a} & {b} \\ {c} & {-a}\end{array}\right]$$ is the square root of the 2 × 2 identity matrix, then find the relation a between a, b and c.

Explaination: 34. Explaination:  35. If A and B are symmetric matrices, then AB -BA is a _______ matrix.

Explaination:
skew symmetric, as (AB – BA)’
= (AB)’ – (BA)’
= B’A’ – A’B’ = BA- AB = -(AB – BA) (∵ A’ =A,B’ = B)

36. If A is a skew symmetric matrix then A² is a ______ .

Explaination: 37. If A and B are two matrices such that their multiplication is defined, then (AB)’ = _____ .

Explaination: B’A’, as result

38. If matrix A = [1 2 3] then find AA’, where A’ is the transpose of matrix A.

Explaination: 39. Explaination: 40. For what value of x, is the matrix a skew symmetric matrix ? [AI 2013]

Explaination: 41. If A and B are symmetric matrices, show that AB is symmetric, if AB = BA.

Explaination: Given, A’ = A, B’ = B and if AB is symmetric, then
(AB)’ =AB … (i)
Also, (AB)’ = B’A’ = BA … (ii)
From (i) and (ii), we get AB = BA.

42. skew symmetric, find the values of ‘a’ and ‘b’ [CBSE 2018]

Explaination: For skew symmetric matrix, aij = – aji
⇒ a = – 2, b = 3.

43. Show that all the elements on the main diagonal of a skew symmetric matrix are zero. [Delhi 2017]

Explaination: A square matrix A = [aij is skew symmetric if
aij = –ji, ∀ i,j
Let i=j
⇒ aii = – aii
⇒ 2aii = 0
=> aii = 0
Hence, all the diagonal elements of a skew symmetric matrix are always zero.

44. If A and B are symmetric matrices, show that AB + BA is symmetric and AB – BA is skew symmetric. [Dehradun 2019]

Explaination: A’ =A,B’= B;
Consider (AB + BA)’ = (AB)’ + (BA)’
= B’A’ + A’B’
= BA+AB
= AB +BA.
Hence, symmetric.
Consider (AB – BA)’ = (AB)’ – (BA)’
= B’ A’- A’ B’
= BA – AB
= -(AB-BA)
Hence, skew symmetric.

45. Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.