Practice Question On Matrix and determinant | CBSE | IITJEE | MAINS | ADVANCED | NDA | AIRFORCE | 31-45

Matrix and determinant

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Question no. 31

If A= , then A¹⁰⁰ =

looks_one 2¹⁰⁰A

looks_two2⁹⁹A

looks_3 2¹⁰¹A

looks_4 None of these

Option looks_two2⁹⁹A

Solution :

Characteristic eqaution :
(1- λ)² - 1 = 0
λ² + 1 - 2λ - 1 = 0
λ² - 2λ = 0
A² = 2A
A³ = 2A² = 2²A
A⁴ = 2A³ = 2³ A
.
.
.
.
A¹⁰⁰ = 2⁹⁹A

Question no. 32

Matrix A= is invertible for

looks_one k=1

looks_two k=0

looks_3 k=-1

looks_4 All real k

option looks_two k=0

Solution :
For Invertible , |A| ≠ 0
1(1) - k(k) ≠ 0
k² ≠ 1
K ≠ 1 , -1

Question no. 33

If , then values of x,y,z,w are

looks_one 2,2,3,4

looks_two 2,3,1,2

looks_3 3,3,0,1

looks_4 None of these

option looks_one 2,2,3,4

Solution :
x+ y = 4 .....[1]
x- y = 0 .......[2]
Solve [1] and [2]
x = 2 , y = 2
2x + z = 7 ⇒ 4 + z = 7 ⇒ z = 3
2z + w = 10 ⇒ 6 + w = 10
w = 4

Question no. 34

If A and B are 3✕3 matrices such that AB=A and BA=B, then

looks_one A²=A and B² ≠ B

looks_two A² ≠ A and B² = B

looks_3 A² = A and B² = B

looks_4 A² ≠ A and B² ≠ B

option looks_3 A² = A and B² = B

Solution :
Case 1 :
AB = A
A(BA) = A
ABA = A
(AB)A = A
AA = A
A² = A
Case 2 :
BA = B
B(AB) = B
BB = B
B² = B

Question no. 35

If I is unit matrix , then 3I will be

looks_one A unit matrix

looks_two A triangular matrix

looks_3 A scalar matrix

looks_4 None of these

option looks_3 A scalar matrix

Solution :
I is identity matrix
whereas 3I will be scalar matrix

Question no. 36

If A= , then A² - 5A =

looks_one I

looks_two 14I

looks_3 0

looks_4 None of these

option looks_two 14I

Solution :

Question no. 37

If then A⁴ =

looks_one

looks_two

looks_3

looks_4

option looks_one

Solution :

Question no. 38

If , then

looks_one AB = O, BA = O

looks_two AB = O, BA ≠ O

looks_3 AB ≠ O, BA = O

looks_4 AB ≠ O, BA ≠ O

Option looks_one AB = O, BA = O

Solution :
|A| = 0
|B| = 0
then
AB = 0
BA = 0

Question no. 39

Choose the correct answer

looks_one Every identity matrix is a scalar matrix

looks_two Every scalar matrix is an identity matrix

looks_3 Every diagonal matrix is an identity matrix

looks_4 A square matrix whose each element is 1 is identity matrix

option looks_one Every identity matrix is a scalar matrix

Solution :
Every identity matrix is a scalar matrix

Question no. 40

If A and B are square matrices of order n ✕ n , then matrices ( A -B )² is equal to

looks_one A² - B²

looks_two A² -2AB + B²

looks_3 A² +2AB + B²

looks_4 A² -AB -BA + B²

Option looks_4 A² -AB -BA + B²

Solution :
( A -B )² = (A-B)(A-B)
(A-B)(A-B) = A² -AB -BA + B²

Question no. 41

If then VU +X'Y =

looks_one 20

looks_two [20]

looks_3 [-20]

looks_4 -20

Option looks_two [20]

Solution :

Question no. 42

Which one of the following is not true

looks_one Matrix addition is commutative

looks_two Matrix addition is associative

looks_3 Matrix multiplication is commutative

looks_4 Matrix multiplication is associative

Option looks_3 Matrix multiplication is commutative

Solution :
Matrix multiplication is not commutative

Question no. 43

If , then

looks_one A² = A

looks_two B² = B

looks_3 AB ≠ BA

looks_4 AB = BA

option looks_3 AB ≠ BA

Solution :

AB ≠ BA

Question no. 44

Which of the following is incorrect?

looks_one A² -B² = (A+B)(A-B)

looks_two (A^T)^T = A

looks_3 (AB)ⁿ= AⁿBⁿ where A, B commute

looks_4 (A-I) (I+A) = O , that imply A² = I

option looks_one A² -B² = (A+B)(A-B)

Solution :
(A+B)(A-B) = A² + BA -AB -B²
BA ≠ AB
A² -B² ≠ (A+B)(A-B)

Question no. 45

If , then X =

looks_one

looks_two

looks_3

looks_4

Option looks_one

Solution :
AX = B
X = A^-1B

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