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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

**Answer/Explanation**

Answer: b

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

2. If A = [a_{ij}] is a 2 × 3 matrix, such that a_{ij} = \(\frac{(-i+2 j)^{2}}{5}\).then a_{23} is

**Answer/Explanation**

Answer: d

Explaination: (d), as a_{23} = \(\frac{(-2+6)^{2}}{5}=\frac{16}{5}\)

3. If A = diag(3, -1), then matrix A is

**Answer/Explanation**

Answer: c

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

**Answer/Explanation**

Answer: d

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

**Answer/Explanation**

Answer: 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

**Answer/Explanation**

Answer: b

Explaination: (b), by definition.

7.

**Answer/Explanation**

Answer: d

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

**Answer/Explanation**

Answer: a

Explaination:

(a), as in skew symmetric matrix, a_{ij} = -a_{ji}

⇒ a_{ii} = – a_{ii}

⇒ 2a_{ii} = 0

⇒ a_{ii} = 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

**Answer/Explanation**

Answer: b

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.

**Answer/Explanation**

Answer: c

Explaination: (c), as it satisfies a_{ij} = a_{ji}, = -a_{ji} and a_{ii} = 0

11.

**Answer/Explanation**

Answer:

Explaination: False, as their orders are different.

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

**Answer/Explanation**

Answer:

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

13. If A is a 3 × 3 matrix, whose elements are given by ai_{ij} = \(\frac{1}{3}\)|-3i + j|, then write the value a_{23}. [Foreign 2013]

**Answer/Explanation**

Answer:

Explaination: Given a_{ij} = \(\frac{1}{3}\)|-3i + j|

∴ a_{23} = \(\frac{1}{3}\)|-6 + 3| = 1.

14.

**Answer/Explanation**

Answer:

Explaination:

15.

**Answer/Explanation**

Answer:

Explaination: x + y + z = 9, x + z = 5, y + z = 1, on solving we get

x = 2, y = 4, z = 3

16.

**Answer/Explanation**

Answer:

Explaination:

cos α = 1, sin α = 0

⇒ α = 0°

17.

**Answer/Explanation**

Answer:

Explaination: 3a_{22} – 4a_{33} = 3 × 5 – 4 × 4 = -1

18.

**Answer/Explanation**

Answer:

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

19.

**Answer/Explanation**

Answer:

Explaination:

20. If matrix X is such that

then order of matrix X is ______ .

**Answer/Explanation**

Answer:

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 _____ .

**Answer/Explanation**

Answer:

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.

**Answer/Explanation**

Answer:

Explaination: False, as AB ≠ BA in general.

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

**Answer/Explanation**

Answer:

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

24.

**Answer/Explanation**

Answer:

Explaination: a_{22} + b_{21} = 4 – 3 = 1

25.

then write the value of k.

**Answer/Explanation**

Answer:

Explaination:

26.

**Answer/Explanation**

Answer:

Explaination:

27. Find x and y, if

**Answer/Explanation**

Answer:

Explaination:

28.

find the value of x [Delhi 2012]

**Answer/Explanation**

Answer:

Explaination:

29.

**Answer/Explanation**

Answer:

Explaination:

30. Find a matrix X such that 2A + B + X =

**Answer/Explanation**

Answer:

Explaination:

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

**Answer/Explanation**

Answer:

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

32.

**Answer/Explanation**

Answer:

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.

**Answer/Explanation**

Answer:

Explaination:

34.

**Answer/Explanation**

Answer:

Explaination:

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

**Answer/Explanation**

Answer:

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 ______ .

**Answer/Explanation**

Answer:

Explaination:

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

**Answer/Explanation**

Answer:

Explaination: B’A’, as result

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

**Answer/Explanation**

Answer:

Explaination:

39.

**Answer/Explanation**

Answer:

Explaination:

40. For what value of x, is the matrix

a skew symmetric matrix ? [AI 2013]

**Answer/Explanation**

Answer:

Explaination:

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

**Answer/Explanation**

Answer:

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]

**Answer/Explanation**

Answer:

Explaination: For skew symmetric matrix, a_{ij} = – a_{ji}

⇒ a = – 2, b = 3.

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

**Answer/Explanation**

Answer:

Explaination: A square matrix A = [a_{ij} is skew symmetric if

a_{ij} = –_{ji}, ∀ i,j

Let i=j

⇒ a_{ii} = – a_{ii}

⇒ 2a_{ii} = 0

=> a_{ii} = 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]

**Answer/Explanation**

Answer:

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.

**Answer/Explanation**

Answer:

Explaination: Let A is symmetric then A’ = A …(i)

Now (B’AB)’ = B’A'(B’)’ = B’A’B = B’AB [using (i)]

Hence, symmetric.

Similarly, let A be skew symmetric then A’= -A

(B’AB)’ = B’A'(B’)’ = B’A’B

= B'(-A)B = -B’AB.

Hence, B’AB is skew symmetric.

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