Determinants Test

Result

Determinants Test - 7

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Weekly Quiz Competition

Question 1
1 / -0.25

A
–1

B
1

C
0

D
none of these.

Solution
Question 2
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The only integral root of the equation det. is

A
3

B
2

C
4

D
1

Solution

Clearly , y = 1 satisfies it. [C₃ = 3C₁ ]
Question 3
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If 1/a+1/b+ 1/c = 0 , then

A
abc

B
0

C
–abc

D
none of these

Solution

Since ,
Question 4
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If A is a non singular matrix and A ’denotes the transpose of A , then

A
|A|+|A ′| ≠0

B
|A| ≠|A ′|

C
|AA ′| ≠|A² |

D
none of these

Solution

Because , |A|=|A ′|
Question 5
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If the matrix AB = O , then

A
A = O or B = O

B
A = O and B = O

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

D
none of these.

Solution

If the matrix AB = O , then , marix A can be a non zero matrix as well as matrix B can be a non zero matrix. Which means det.A = 0 and det.B = 0.
Question 6
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If A ,B andC be the three square matrices such that A = B + C , then Det A is equal to

A
det B

B
det C

C
det B + det C

D
none of these

Solution

Because , |A| ≠|B|+|C|
Question 7
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Solution set of the equation

A
{–3 , 1, 5}

B
{2 , 1, 5}

C
{2 , 0 ,1}

D
{2 , –3 ,1}

Solution

[x(-3x(x+2) - 2x(x-3)]+6[2(x+2)+3(x-3)]-1(4x-9x) = 0

⇒- 5x³ + 35x - 30 = 0
⇒(x-1)(x-2)(x+3) = 0 ⇒x=1,2,-3
Question 8
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A determinant is unaltered , if

A
Every element in a column is multiplied by the same factor

B
Two rows are interchanged

C
Two columns are interchanged

D
To each element of any row is added the corresponding element of the other row multiplied by a given factor

Solution

This is because of the elementary transformations of determinants . The value of determinant remains unaffected by applying elementary transformations.
Question 9
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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
–9

C
–27

D
-81

Solution

|3AB| = 27|A||B| = 27(-1)(3) = -81
Question 10
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A
1

B
0

C
abc

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 11
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If 1 , ω,ω² are cube roots of unity , then has value

A
0

B
1

C
–1

D
none of these.

Solution

write 1 as in R₁ and take out common from R₁ , we get :

because row 1 and row 3 are identical.
Question 12
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Solution set of the equation

A
{1, 2 ,7}

B
{–9 ,2 , 7}

C
{2 , 5, 6}

D
none of these

Solution

x(x² - 12) - 3(2x - 14)+7(12 - 7x) = 0

⇒x³ - 67x + 126 = 0

⇒(x-2)(x-7)(x+9) = 0 ⇒x = 2,7,-9
Question 13
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A
3k

B
2k

C
6/k

D
6k

Solution
Question 14
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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 , (adj A) B ≠ O

D
if det. A = 0 , (adj A) B ≠ O

Solution

Explanation here 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 15
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If A is a square matrix such that A³ = I , then A⁻¹ is equal to

A
I

B
A²

C
A

D
none of these.

Solution

A³ = I ⇒ Pre - multiplying both sides by A⁻¹ ,A⁻¹ , A³ = A⁻¹ I ⇒A² = A⁻¹