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QUESTION: 1

Solution:

⇒ (x-y)(y-z)[y^{2} + yz+z^{2} - x^{2} - xy - y^{2}]

⇒ (x-y)(y-z) (z-x) (x+y+z)

QUESTION: 2

If , then is equal to

Solution:

=1(-5) -2(10) + 3(11)

=-5-20+33 = 8

=1(-30) - 6(20) + 3(66)

= -30-120+198= 48

D = 8 ⇒ 6D = 48

QUESTION: 3

Solution:

Apply, R_{2} →R_{2} - R_{1},

Apply, R_{3} →R_{3} - R_{1},

⇒ (x-3) (6x -9) = 0 ⇒x =

QUESTION: 4

If the order of matrix A is m*p. And the order of B is p×n. Then the order of matrix AB is ?

Solution:

QUESTION: 5

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

Solution:

QUESTION: 6

One root of the equation

Solution:

Apply , R_{1}→R_{1} +R_{2}+R_{3},

⇒either (3x -2) = 0

QUESTION: 7

Solution:

Apply , C_{1} → C_{1}+C_{2}+C_{3}+C_{4}

Apply, R1 → R_{1} - R_{2},

Apply, R_{1}→R_{1} - R_{2}

⇒ (10+ x) x^{3}

QUESTION: 8

The determinant is equal to

Solution:

Apply, C_{1} → C_{1 }+ C_{2} + C_{3},

QUESTION: 9

Solution:

The determinant of a lower triangular (or an upper triangular matrix is equal to the product of the diagonal elements.

QUESTION: 10

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

Solution:

where n is order of matrix. Here n = 3.

QUESTION: 11

Solution:

QUESTION: 12

The only integral root of the equation det. is

Solution:

Clearly , y = 1 satisfies it. [C_{3} = 3C_{1}]

QUESTION: 13

If 1/a+1/b+ 1/c = 0 , then

Solution:

Since ,

QUESTION: 14

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

Solution:

Because , |A|=|A′|

QUESTION: 15

If the matrix AB = O , then

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: 16

If A ,B andC be the three square matrices such that A = B + C , then Det A is equal to

Solution:

Because , |A| ≠ |B|+|C|

QUESTION: 17

Solution set of the equation

Solution:

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

⇒ - 5x^{3}_{ }+ 35x - 30 = 0

⇒ (x-1)(x-2)(x+3) = 0 ⇒ x=1,2,-3

QUESTION: 18

A determinant is unaltered , if

Solution:

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

QUESTION: 19

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

Solution:

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

QUESTION: 20

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: 21

If 1 , ω,ω^{2} are cube roots of unity , then has value

Solution:

write 1 as in R_{1} and take out common from R_{1} , we get :

because row 1 and row 3 are identical.

QUESTION: 22

Solution set of the equation

Solution:

x(x^{2} - 12) - 3(2x - 14)+7(12 - 7x) = 0

⇒ x^{3} - 67x + 126 = 0

⇒(x-2)(x-7)(x+9) = 0 ⇒x = 2,7,-9

QUESTION: 23

Solution:

QUESTION: 24

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

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: 25

If A is a square matrix such that A^{3} = I , then A^{−1} is equal to

Solution:

A^{3} = I ⇒ Pre - multiplying both sides by A^{−1},A^{−1}, A^{3} = A^{−1} I ⇒ A^{2} = A^{−1}

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