Determinants Test

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Determinants Test - 9

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Question 1
1 / -0

If A is a square matrix such that A² = A, then, det.(A) = ………

A
2 or -2

B
0 or 1

C
1 or - 1

D
None of these

Solution

Since, A² = A ⇒ |A|² = |A|

∴ (Det. A)(Det. A) = Det.A, ∴ det.A [ det.A - 1 ] = 0. ∴ det.A = 0 or 1.
Question 2
1 / -0

If A’ is the transpose of a square matrix A, then

A
|A| + |A'| = 0

B
|A| ≠ |A'|

C
|A| = |A'|

D
None of these

Solution

The determinant of a matrix A and its transpose always same. Because if we interchange the rows into column in a determinant the value of determinant remains unaltered.
Question 3
1 / -0

Let A and B be 3 × 3 matrices, then AB =O implies

A
|A| = 0 & |B| = 0

B
A =O or B = O

C
A=O & B = O

D
either |A| = 0 or |B| = 0

Solution

if AB=0 Then|AB|=0 and if |AB| = 0 ⇒ either |A| = 0 or |B| = 0
Question 4
1 / -0

If A and B are any 2 × 2 matrices , then det. (A+B) = 0 implies

A
det A = 0 and det B = 0

B
det A = 0 or det B = 0

C
detA + det B = 0

D
None of these

Solution

If det (A+B)=0 implies that A+B a Singular matrix.
Question 5
1 / -0

If A and B are square matrices of same order and A’ denotes the transpose of A, then

A
(AB)’ = A’B’

B
AB = O ⇒ A = 0 or B = 0

C
AB = O ⇒|A| = 0 and |B| = 0

D
(AB)’ = B’A’

Solution

By the property of transpose of a matrix, (AB)’ = B’A’.
Question 6
1 / -0

In a third order determinant, each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of the third column consists of sum of four terms. Then it can be decomposed into n determinants, where n has value

A
9

B
16

C
1

D
24

Solution

N = 2 ×3 × 4 = 24.
Question 7
1 / -0

If each element of a 3 × 3 matrix A is multiplied by 3, then the determinant of the newly formed matrix is

A
9 det A

B
27 det A

C
3 det A

D
( det det A)³

Solution

|3A| = 3³|A| = 27|A|

if A is a square matrix of order n, then |kA|=kⁿ |A| where n is the order of matrix
Question 8
1 / -0

If A is a non singular matrix of order 3, then |adj(adj A)|

A
|A|³

B
|A|⁴

C
|A|⁶

D
None of these

Solution

|adj.(adj.A) = |A|ⁿ⁺¹, where n is order of matrix. Here n = 3.
Question 9
1 / -0

If the matrix AB = O , then

A
It is not necessary that either A = O or B = O

B
A = O or B = O

C
A = O and B = O

D
None of these

Solution

If the matrix AB = O , then , matrix A can be a non zero matrix as well as matrix B can be a non zero matrix because for the multiplication of two matrics to be equal to 0 the matrices need not to be equal to 0. So, it is not necessary that either A=0 or B=0.

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Determinants Test - 9

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Determinants Test - 9

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SHARING IS CARING

Question 1

2 or -2

0 or 1

1 or - 1

None of these

Solution

Question 2

|A| + |A'| = 0

|A| ≠ |A'|

|A| = |A'|

None of these

Solution

Question 3

|A| = 0 & |B| = 0

A =O or B = O

A=O & B = O

either |A| = 0 or |B| = 0

Solution

Question 4

det A = 0 and det B = 0

det A = 0 or det B = 0

detA + det B = 0

None of these

Solution

Question 5

(AB)’ = A’B’

AB = O ⇒ A = 0 or B = 0

AB = O ⇒|A| = 0 and |B| = 0

(AB)’ = B’A’

Solution

Question 6

9

16

1

24

Solution

Question 7

9 det A

27 det A

3 det A

( det det A)3

Solution

Question 8

|A|3

|A|4

|A|6

None of these

Solution

Question 9

It is not necessary that either A = O or B = O

A = O or B = O

A = O and B = O

None of these

Solution

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( det det A)³

|A|³

|A|⁴

|A|⁶