Determinants Test

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Determinants Test - 8

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Weekly Quiz Competition

Question 1
1 / -0

If A is a non singular matrix and A’ denotes the transpose of A, then

A
|AA'| ≠|A²|

B
|A|≠|A'|

C
|A| + |A'| ≠ 0

D
None of these

Solution

Because, the determinant of a matrix and its transpose are always equal that is |A| = |A'|
Question 2
1 / -0

If A and B are square matrices of order 3, such that Det.A = –1, Det.B =3 then, the determinant of 3AB is equal to

A
-81

B
–27

C
–9

D
81

Solution

|3AB| = 3³|A||B| = 27(-1)(3) = -81
Question 3
1 / -0

A square matrix A is called singular if det. A is

A
Positive

B
Negative

C
0

D
non–zero

Solution

For a singular matrix, |A| = 0.
Question 4
1 / -0

The system AX = B of n equations in n unknowns has infinitely many solutions if

A
det. A≠0

B
if det.A = 0, ( adj A)B≠O

C
if det. A≠ 0, (adjA)B≠ O

D
if det. A = 0 , ( adj A ) B =O

Solution

If det. A = 0, (adj A) B =O⇒ The system AX = B of n equations in n unknowns may be consistent with infinitely many solutions or it may be inconsistent.
Question 5
1 / -0

Let A be a skew – symmetric matrix of order n then

A
|A| = 0 if n is odd

B
|A| = 0 for all n ∈N

C
|A| = 0 if n is even

D
None of these

Solution

Because, the determinant of a skew symmetric matrix of odd order is always zero and of even order is a non zero perfect square.
Question 6
1 / -0

For an invertible square matrix of order 3 with real entries A^-1 = A², then det. A =

A
1/3

B
3

C
1

D
None of these

Solution

A² = I ⇒ A²A^-1 = IA^-1 ⇒ A = A^-1 and it is possible only if A is an identity matrix and determinant of the identity matrix is equal to 1
Question 7
1 / -0

The value of the determinant of a skew symmetric matrix of even order is

A
Negative

B
None of these

C
0

D
A non zero perfect square

Solution

The determinant of a skew symmetric matrix of even order is A non zero perfect square and odd order is equal to 0.

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Determinants Test - 8

Result

Determinants Test - 8

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Question 1

|AA'| ≠|A2|

|A|≠|A'|

|A| + |A'| ≠ 0

None of these

Solution

Question 2

-81

–27

–9

81

Solution

Question 3

Positive

Negative

0

non–zero

Solution

Question 4

det. A≠0

if det.A = 0, ( adj A)B≠O

if det. A≠ 0, (adjA)B≠ O

if det. A = 0 , ( adj A ) B =O

Solution

Question 5

|A| = 0 if n is odd

|A| = 0 for all n ∈N

|A| = 0 if n is even

None of these

Solution

Question 6

1/3

3

1

None of these

Solution

Question 7

Negative

None of these

0

A non zero perfect square

Solution

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|AA'| ≠|A²|