MP Board Class 12th Maths Important Questions Chapter 3 Matrices

MP Board Class 12th Maths Important Questions Chapter 3 Matrices

MP Board Class 12th Maths Important Questions Chapter 3 Matrices

Important Questions

Short Answer Type Questions

Question 3.
If A = and B =  then find 3A – 5B? (NCERT)
Solution:

Question 4.

Question 5.
From the following equation find the value of x and y?

From defnition of matrix,
2x + 3 = 7
⇒ 2x = 4 ⇒ x = 2
⇒ 2y – 4 = 14
⇒ 2y = 18 ⇒ y = 9
∴ x = 2, y = 9.

Question 7.
Find the value of x and y

By defnition of matrix,
2 + y = 5 ⇒ y = 3
2x + 2 = 8
⇒ x + 1 = 4
⇒ x = 3
∴ x = 3, y = 3.

Question 8.

By defnition of matrix,
x + y + z = 9
x + z = 5
y + z = 7
From eqns. (1) and (2),
x + y + z = 9
⇒ 5 + y = 9 ⇒ y = 4
From eqns. (1) and (3),
x + (y + z) = 9
⇒ x + 7 = 9
⇒ x = 2
Putting the value of x in eqn. (2),
2 + z = 5
⇒ z = 3
∴ x = 2, y = 4, z = 3.

Question 9.

By defnition of matrix,
x + y = 6 ……………….. (1)
xy = 8 ………………. (2)
5 + z = 5
⇒ z = 0
From eqn. (1), y = 6 – x
xy = 8
⇒ 6x – x² = 8
⇒ x² – 6x + 8 = 0
⇒ x² – 4x – 2x + 8 = 0
⇒ x (x – 4) – 2 ( x – 4) = 0
⇒ ( x – 2) (x – 4) = 0
⇒ x = 2, 4
When x = 2 then y = 6 – 2 = 4
When x = 4 then y = 6 – 4 = 2
So x = 2, y = 4, z = 0
x = 4, y = 2, z = 0.

Question 16.

Question 17.

Question 21.

From eqns. (1) and (2),
(AB)’ = B’A’. proved.

Long Answer Type Questions

Question 1.

⇒ (adj A).A = |A| I
From eqns. (1) and (2),
A.adj A = (adj A) .A = |A| I. Proved.

Question 3.

A₁₁ = (-1)² (0 – 0) = 0
A₁₂ = (-1)³ (0 – 0) = 0
A₁₃ = (-1)⁴ ( 0 – 1) = -1, A₂₁ = (-1)³ (0 – 0) = 0
A₂₂ = (-1)⁴ ( 0 – 1) = -1, A₂₃ = (-1)⁵ (0 – 0) = 0
A₃₁ = (-1)⁴ ( 0 – 1) = -1, A₃₂ = (-1)⁵ (0 – 0) = 0
A₃₃ = (-1)⁶ ( 0 – 0) = 0

Question 4.

Question 6.

⇒ |A| = 1 (1 – 4) + 2 ( 4 – 2) + 2 ( 4 – 2)
= – 3 + 4 + 4 = 5
If |A| ≠ 0, then A^-1 exists

Question 8.

∴ A₁₁ = (-1)¹⁺¹ 4 = 4
A₁₂ = (-1)¹⁺² (3) = -3
A₂₁ = (-1)²⁺¹ (-3) = 3
A₂₂ = (-1)²⁺² (2) = 2.

Question 10.
(A) Solve the following equations by matrix method:
x + y + z = 3
2x – y + z = 2
x – 2y + 3z = 2.
Solution:

(B) Solve the following equations by matrix method:
x + y + z = 6
x + 2y = 3z = 14
x + 4y + 9z = 36
Solution:
x + y + z = 6
x + 2y + 3z = 14
x + 4y + 9z = 36.
Where

X = A^-1 B

∴ x = 1, y = 2, z = 3.

Question 11.

(i) (A + B)’ = A’ + B’
(ii) (A – B)’ = A’ – B’. (NCERT)
Solution:
(i) Given

Question 12.

(i) (A + B)’ = A’ + B’
(ii) (A – B)’ = A’ – B’. (NCERT)
Solution:
solve like Q.No.11.

Question 13.
Express matrix A =  as sum of a symmetric and a skew symmetric matrix? (NCERT)
Solution:

Eqn. (1) and symmetric matrix and eqn. (2) is skew symmetric matrix.

Question 14.
(A) By using elementary operations, find the inverse of matrix A =
Solution:
Using A = AI

(B) By using elementary operation, find the inverse of matrix A =
Solution:
Solve like Q.No. 14 (A).
Answer:

Question 15.
Solve the following system of equations by using matrix method: (NCERT, CBSE 2011)

⇒ |A| = 2 × (120 – 45) -3 (- 80 – 30) + 10 (36 + 36)
⇒ |A| = 150 + 330 + 720 = 1200
⇒ |A| ≠ 0, hence A^-1 exists.
Applying formula
X = A^-1B

Question 16.

x + 2y – 3z = -4
2x + 3y + 2z = 2
3x – 3y – 4z = 11. (CBSE 2008, 10, 12)
Solution:
Given:

⇒ |A| = 1(- 12 + 6) – 2 (- 8 – 6) – 3 (- 6 – 9)
⇒ |A| = 1 (-12 + 6) -2 (-8 -6) -3 (-6 -9)
⇒ |A| ≠ 0
⇒ Hence A^-1 exists.
Hence



Equation of above matrix
x + 2y – 3z = – 4
2x + 3y + 2z = 2
3x – 3y – 4z = 11
Solution of above equations
AX = B

Question 17.
The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs. 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs. 90. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is Rs. 70. Find the cost of each item per kg by matrix method?
Solution:
Let the cost of 1 kg onion = Rs. x
1 kg wheat = Rs. y
and 1 kg rice = Rs. z
According to equation
4x + 3y + 2z = 60
2x + 4y + 6z = 90
6x + 2y + 3z = 70
Matrix form will be
AX = B
Where

⇒ Hence A^-1 exist.
Applying formula
X = A^-1 B



∴ x = Rs. 5, y = Rs. 8, z = Rs. 8.

Objective Type Questions:

Question 1.

Question 2.
If A =  , then A⁵ is equal to:
(a) 5 A
(b) 10 A
(c) 16 A
(d) 32 A
Answer:
(c) 16 A

Question 3.
If a matrix is both symmetric and skew – symmetric, then:
(a) A is a diagonal matrix
(b) A is zero matrix
(c) A is a square matrix
(d) None of these
Answer:
(b) A is zero matrix

Question 2.
Fill in the blanks:

1. 
2. If A = diag [1, -1, 2] and B = diag [2, 3, -1], then value of 3A + 4B will be ……………………
3. A matrix A is said to be idempotent if ………………………
4. A matrix A is said to be orthogonal if ……………………..
5. 

Answer:

1. 
2. diag [11, 9, 2]
3. A² = A
4. AA’ = A’A = I
5. x = 2

Question 3.
Write True/False:

1. Multiplication of matrix is always commutative?
2. Two matrix are said to be comparable if they have same number of rows and columns?
3. If A is a square matrix, then A. adj A = |A| I?
4. A square matrix A is said to be symmetric if A = – A^T?
5. Matrix A and B are inverse of each other if AB = BA?

Answer:

1. False
2. True
3. True
4. False
5. False

Question 4.
Match the Column:

Answer:

1. (d)
2. (e)
3. (a)
4. (b)
5. (c)

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