Show that [{:(,a(1)l(1)+b(1)m(1),a(1)l(2...

Show that `[{:(,a_(1)l_(1)+b_(1)m_(1),a_(1)l_(2)+b_(1)m_(2),a_(1)l_(3)+b_(1)m_(3)),(,a_(2)l_(1)+b_(2)m_(1),+a_(2)l_(2)+b_(2)m_(2),a_(2)l_(3)+b_(2)m_(3)),(,a_(2)l_(1)+b_(3)m_(1),+a_(3)l_(2)+b_(2)m_(2),a_(3)l_(3)+b_(3)m_(3))]:}`

`a_(1) a_(2) a_(3) b_(1) b_(2) b_(3)`

`x_(1) x_(2) x_(3) y_(1) y_(2) y_(3)`

none of these

Text Solution

Verified by Experts

The correct Answer is:

We have,
`|(a_(1),b_(1),0),(a_(2),b_(2),0),(a_(3),b_(3),0)||(x_(1),y_(1),0),(x_(2),y_(2),0),(x_(3),y^(3),0)| = 0 xx 0 = 0`
`rArr |(a_(1) x_(1) + b_(1) y_(1) + 0,a_(1) x_(2) + b_(1) y_(2) + 0,a_(1) x_(3) + b_(1) y_(3) + 0),(a_(2) x_(1) + b_(2) y_(1) + 0,a_(2) x_(2) + b_(2) y_(2) + 0,a_(2) x_(3) + b_(2) y_(3) + 0),(a_(3) x_(1) + b_(3) y_(1) + 0,a_(3) x_(2) + b_(3) y_(2) + 0,a_(3) x_(3) + b_(3) y_(3) + 0)| = 0` [Using row by-row multiplication]
`rArr |(a_(1) x_(1) + b_(1) y_(1),a_(1) x_(2) + b_(1) y_(2),a_(1) x_(3) + b_(1) y_(3)),(a_(2) x_(1) + b_(2) y_(1),a_(2) x_(2) + b_(2) y_(2),a_(2) x_(3) + b_(2) y_(3)),(a_(3) x_(1) + b_(3) y_(1),a_(3) x_(2) + b_(3) y_(2),a_(3) x_(3) + b_(3) y_(3))| = 0`

Topper's Solved these Questions

DETERMINANTS
OBJECTIVE RD SHARMA ENGLISH|Exercise Section I - Solved Mcqs|74 Videos
DETERMINANTS
OBJECTIVE RD SHARMA ENGLISH|Exercise Section II - Assertion Reason Type|6 Videos
CARTESIAN CO-ORDINATE SYSTEM
OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|27 Videos
DISCRETE PROBABILITY DISTRIBUTIONS
OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|40 Videos

Similar Questions

Explore conceptually related problems

The value of |(a_(1) x_(1) + b_(1) y_(1),a_(1) x_(2) + b_(1) y_(2),a_(1) x_(3) + b_(1) y_(3)),(a_(2) x_(1) +b_(2) y_(1),a_(2) x_(2) + b_(2) y_(2),a_(2) x_(3) + b_(2) y_(3)),(a_(3) x_(1) + b_(3) y_(1),a_(3) x_(2) + b_(3) y_(2),a_(3) x_(3) + b_(3) y_(3))| , is

Prove that |{:(a_(1)alpha_(1)+b_(1)beta_(1),a_(1)alpha_(2)+b_(1)beta_(2),a_(1)alpha_(3)+b_(1)beta_(3)),(a_(2)alpha_(1)+b_(2)beta_(1),a_(2)alpha_(2)+b_(2)beta_(2),a_(2)alpha_(3)+b_(2)beta_(3)),(a_(3)alpha_(1)+b_(3)beta_(1),a_(3)alpha_(2)+b_(3)beta_(2),a_(3)alpha_(3)+b_(3)beta_(3)):}| =0.

Show that if x_(1),x_(2),x_(3) ne 0 |{:(x_(1) +a_(1)b_(1),,a_(1)b_(2),,a_(1)b_(3)),(a_(2)b_(1),,x_(2)+a_(2)b_(2),,a_(2)b_(3)),(a_(3)b_(1),,a_(3)b_(2),,x_(3)+a_(3)b_(3)):}| =x_(1)x_(2)x_(3) (1+(a_(1)b_(1))/(x_(1))+(a_(2)b_(2))/(x_(2))+(a_(3)b_(3))/(x_(3)))

If |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =5 , then the value of Delta = |(b_(2) c_(3) - b_(3) c_(2),a_(3) c_(2) - a_(2) c_(3),a_(2) b_(3) -a_(3) b_(2)),(b_(3) c_(1) - b_(1) c_(3),a_(1) c_(3) - a_(3) c_(1),a_(3) b_(1) - a_(1) b_(3)),(b_(1) c_(2) - b_(2) c_(1),a_(2) c_(1) - a_(1) c_(2),a_(1) b_(2) - a_(2) b_(1))| is

The value of the determinant Delta = |((1 - a_(1)^(3) b_(1)^(3))/(1 - a_(1) b_(1)),(1 - a_(1)^(3) b_(2)^(3))/(1 - a_(1) b_(2)),(1 - a_(1)^(3) b_(3)^(3))/(1 - a_(1) b_(3))),((1 - a_(2)^(3) b_(1)^(3))/(1 - a_(2) b_(1)),(1 - a_(2)^(3) b_(2)^(3))/(1 - a_(2) b_(2)),(1 - a_(2)^(3) b_(3)^(3))/(1 - a_(2) b_(3))),((1 - a_(3)^(3) b_(1)^(3))/(1 - a_(3) b_(1)),(1 - a_(3)^(3) b_(2)^(3))/(1 - a_(3) b_(2)),(1 - a_(3)^(3) b_(3)^(3))/(1 - a_(3) b_(3)))| , is

the value of the determinant |{:((a_(1)-b_(1))^(2),,(a_(1)-b_(2))^(2),,(a_(1)-b_(3))^(2),,(a_(1)-b_(4))^(2)),((a_(2)-b_(1))^(2),,(a_(2)-b_(2))^(2) ,,(a_(2)-b_(3))^(2),,(a_(2)-b_(4))^(2)),((a_(3)-b_(1))^(2),,(a_(3)-b_(2))^(2),,(a_(3)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(4)-b_(1))^(2),,(a_(4)-b_(2))^(2),,(a_(4)-b_(3))^(2),,(a_(4)-b_(4))^(2)):}| is

The determinant |(b_(1)+c_(1),c_(1)+a_(1),a_(1)+b_(1)),(b_(2)+c_(2),c_(2)+a_(2),a_(2)+b_(2)),(b_(3)+c_(3),c_(3)+a_(3),a_(3)+b_(3))|

If omega is a non-real cube root of unity, then Delta = |(a_(1) + b_(1) omega,a_(1) omega^(2) + b_(1),a_(1) + b_(1) + c_(1) omega^(2)),(a_(2) + b_(2) omega,a_(2) omega^(2) + b_(2),a_(2) + b_(2) + c_(2) omega^(2)),(a_(3) + b_(3) omega,a_(3) omega^(2) + b_(3),a_(3) + b_(3) + c_(3) omega^(2))| is equal to

Find the coefficient of x in the determinant |{:((1+x)^(a_(1)b_(1)),(1+x)^(a_(1)b_(2)),(1+x)^(a_(1)b_(3))),((1+x)^(a_(2)b_(1)),(1+x)^(a_(2)b_(2)),(1+x)^(a_(2)b_(3))),((1+x)^(a_(3)b_(1)),(1+x)^(a_(3)b_(2)),(1+x)^(a_(3)b_(3))):}|

OBJECTIVE RD SHARMA ENGLISH-DETERMINANTS-Chapter Test

Show that [{:(,a(1)l(1)+b(1)m(1),a(1)l(2)+b(1)m(2),a(1)l(3)+b(1)m(3)),...
Text Solution
|
STATEMENT-1: The lines a(1)x+b(1)y+c(1)=0a(2)x+b(2)y+c(2)=0,a(3)x+b(3)...
Text Solution
|
If D =|{:(1,1,1),(1,1+x,1),(1,1,1+y):}|"for" " "xne0,yne0 then D is
Text Solution
|
Use properties of determinants to solve for x: |{:(,x+a,b,c),(,c,x+b,a...
Text Solution
|
|(sin^(2) x,cos^(2) x,1),(cos^(2) x,sin^(2) x,1),(- 10,12,2)| =
Text Solution
|
The system of linear equations x + y + z = 2 2x + y -z = 3 3x + ...
Text Solution
|
The roots of the equation |(3x^(2),x^(2) + x cos theta + cos^(2) the...
Text Solution
|
|(bc,bc'+b'c,b'c'),(ca,ca'+c'a,c'a'),(ab,ab'+a'b,a'b')| is equal to
Text Solution
|
If alpha, beta, gamma are the cube roots of 8 , then |(alpha,beta,gamm...
Text Solution
|
One root of the equation |(3x-8, 3, 3),(3,3x-8, 3),(3,3,3x-8)|=0 is ...
Text Solution
|
If a,b and c are non- zero real number then prove that |{:(b^(2...
Text Solution
|
If x, y , z are in A.P., then the value of the det (A) is , where A ...
Text Solution
|
The value of |(b +c,a,a),(b,c +a,b),(c,c,a +b)|, is
Text Solution
|
If a ,\ b ,\ c are non-zero real numbers and if the system of equat...
Text Solution
|
If a!=6,b,c satisfy|[a,2b,2c],[3,b,c],[4,a,b]|=0 ,then abc =
Text Solution
|
The value of Delta = |(1^(2),2^(2),3^(2)),(2^(2),3^(2),4^(2)),(3^(2),4...
Text Solution
|
Prove: |a a+b a+2b a+2b a a+b a+b a+2b a|=9(a+b)b^2
Text Solution
|
If all the elements in a square matrix A of order 3 are equal to 1 or ...
Text Solution
|
Sum of real roots of the euation |{:(1,4,20),(1,-2,5),(1,2x,5x^(2)):}|...
Text Solution
|
If f(x)=|{:(sinx,cosx,tanx),(x^(3),x^(2),x),(2x,1,x):}|, then lim(xto0...
Text Solution
|
If A, B and C are the angles of a triangle and |(1,1,1),(1 + sin A,1...
Text Solution
|