Important Questions for CBSE Class 12 Maths Chapter 3

CBSE Class 12 Maths Chapter-3 Important Questions – Free PDF Download

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CBSE Class 12 Mathematics Important Questions Chapter 3 – Matrices

1 Mark Questions

1. If a matrix has 8 elements, what are the possible orders it can have.
Ans.

2. Identity matrix of orders n is denoted by.
Ans. I_n

3. Define square matrix
Ans. A matrix in which the no. of rows are equal to no. of columns i.e. m = n

4. The no. of all possible metrics of order 3 3 with each entry 0 or 1 is
Ans. 512=2⁹

5.
Write (1) a₃₃, a₁₂ (ii) what is its order
Ans. (i) a₃₃ = 9, a₁₂ = 4
(ii) 3 3

6. Two matrices A = [aij] and B = [bij] are said to be equal if
Ans. They are of the same order.

7. Define Diagonal matrix
Ans. A square matrix in which every non – diagonal element is zero is called diagonal matrix.

8. Every diagonal element of a skew symmetric matrix is
Ans. Zero.

9. If Find
Ans.

10.
Ans.

11. If and A² = I.
Find relation given by a²=I.
Ans.

ATQ.

12. If the matrix A is both symmetric and skews symmetric, then A will be.
Ans. A¹ = A
A¹ = -A
A = -A
2A = 0
A = 0

13. Matrices A and B will be inverse of each other only if
Ans. AB = BA = I

14. If A, B are symmetric matrices of same order, them AB – BA is a
Ans. P = AB – BA

15. Diagonal of skew symmetric matrix are
Ans. Zero

16. If A and B are symmetric matrices of the same order, prove that AB + BA is symmetric
Ans. Let

17. If
Prove that A – A^t is a skew – symmetric matrix
Ans.

Prove

18. If A is any square matrix, prove that AA¹ is symmetric
Ans. Let

19. Solve for x given that
Ans.
2x – 3y = 1
x + y = 3
x = 3 – y
2 (3 – y) – 3y = 1
-5y = -5
y = 1
x = 3 – 1
x = 2

20. Give example of matrices such that AB = 0, BA = 0, A 0, B0
Ans.

21. Show that , is skew symmetric matrix.
Ans.

22. Prove that
is a symmetric matrix
Ans.

23. If show that
Ans.

Prove

24. Solve for x and y, given that
Ans.
x + 2y = 3
3y + 2x = 5
2x + 4y = 6
2x + 3y = 5
y =1
x + 2 (1) = 3
x = 1

25. Given an example of matrix A and B such that AB = 0 but A 0, B 0
Ans.

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4 Marks Questions

1. Find x and y if x + y =and x – y =
Ans.

2.
Show that f(x). f(y) = f(x+y)
Ans. L.H.S = f(x). f(y)

3. If
Find K.So that A² = KA – 2I
Ans. A² = A.A

4.
Prove
Ans.

5.

Ans. Put

L.H.S = R.H.S
Hence prove

6. Construct a 3 4 matrix, whose element are given by aij =
Ans. Let

7. Obtain the inverse of the following matrix using elementary operations

Ans.

8. Let

Find a matrix D such that CD – AB = 0
Ans. Let

2a + 5c – 3 = 0
2b + 5d = 0
3a + 8c – 43 = 0
3b + 8d – 22 = 0
a = -191, b = -110, c = 77, d = 44

9. If , then prove that
where n is any positive integer
Ans. For n = 1

Hence result is true for n =1
Let result is true for n = k

now, we prove their result is true for n = k + 1

P (K + 1) is true Hence P (n) is true.

10. for what values of x
Ans.

4 + 2x + 2x = 0
4x =-4
x = -1

11. Find the matrix X so that
Ans. Let

a = 1, b = -2, c = 2, d = 0

12. , Show that
Where I is the identify matrix of order 2 and n N
Ans. When n = 1

aI + bA = aI + bA
L.H.S = R.H.S
When n = k
(aI + bA)^K = A^KI + Ka^K-1bA……….. (i)
Result is true for n = k
When n = k + 1
(aI + bA)^k+1 = (aI + bA). (aI + bA)^k
= (aI + bA). (a^kI + ka^k-1ba) [From (i)]
= aI (a^kI + ka^k-1ba) + bA (a^kI + ka^k-1 bA)
= a^k+1I + ka^kba + a^kba + ka^k-1 b²A²

= a^k+1+ (k+1) a^kbA
Hence result is true for n = k+1
When eves it is true for n = k

13. Find the values of x, y, z if the matrix
Satisfy the equation = I₃
Ans.

14. If Show that A² – 5A = 7I = 0
Ans. A² – 5A + 7I = 0

A² = 5A – 7I
A² = A².A
= (5A – 7I) .A
= 5A² – 7AI
= 5A² – 7A
= 5(5A – 7I) – 7A
= 25A – 35I – 7A
= 18A – 35I
A⁴ = A³.A
= (18A – 35I).A
=18A² – 35IA
= 18(5A – 7I) – 35A
= 90A – 126I – 35A
= 55A – 126I

15. If A is a square matrix such that A² = A, then (I + A)³ – 7A is equal to
Ans. (I + A)³ – 7A = I³ + A³ + 3IA (I + A) – 7A
= I + A³ + 3I²A + 3IA² – 7A
= I + A³ + 3A + 3A² – 7A
= I + A³ + 3A + 3A – 7A {A² = A}
= I + A³ – A
= I + A² – A
= I + A – A {A² = A}
= I

16. Construct 23 matrix whose element aij are given by

Ans.
For i = j
aij = 4i.j
a₁₁ = 4 1 = 4
a₂₂ = 42 2 = 16
For i < j
aij = 2i + j
a₁₂ = 2 1 +2 = 4
a₁₃ = 2 1 + 3 = 5
a₂₃ = 2 2 + 3 = 7
For i > j
aij = I + 2j
a₂₁ = 2 +2 1 = 4

17. If
then show that A³ – 23A – 40I = 0
Ans.

18. Express the matrix
as the sum of a symmetric and a skew symmetric matrix.
Ans.
Let

is a symmetric matrix
Let

Thus is a skew symmetric matrix

19. If

Ans. For n = 1

Result is true for n = 1
Let it be true for n = k

Thus result is true for n = k+1
Whenever it is true for n = k

20. If
then find the matrix X such that 2A + 3X = 5B.

Ans. 3X = 5B – 2A

21. If then prove that
Ans. For n = 1

Result is true for n = 1
Let result is true for n = k

Thus result is true for n = k + 1
Whenever result is true for n = k

22. Find X and Y, if 2x + 3y =
Ans. On adding

23. If

Show that AB is a zero matrix if and differ by an odd multiple of
Find the condition for which AB=0
Ans.

is odd multiple of