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If x,y,z are nonzero real number , then ...

If x,y,z are nonzero real number , then the inverse of matrix `A= {:[( x,0,0),( 0,y,0) ,( 0,0,z) ]:}` is

A

`{:[(x^(-1), 0,0),( 0,y^(-1) , 0),( 0,0,z^(-1))]:} `

B

`xyz {:[( x^(-1) ,0,0),( 0,y^(-1) ,0),( 0,0,z^(-1)) ]:}`

C

`(1)/(xyz) {:[( x ,0,0),( 0,y ,0),( 0,0,z) ]:}`

D

`(1)/(xyz) {:[( 1 ,0,0),( 0,1 ,0),( 0,0,1) ]:}`

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The correct Answer is:
A
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Let A be an mxxn matrix. If there exists a matrix L of type nxxm such that LA=I_(n) , then L is called left inverse of A. Similarly, if there exists a matrix R of type nxxm such that AR=I_(m) , then R is called right inverse of A. For example, to find right inverse of matrix A=[(1,-1),(1,1),(2,3)] , we take R=[(x,y,x),(u,v,w)] and solve AR=I_(3) , i.e., [(1,-1),(1,1),(2,3)][(x,y,z),(u,v,w)]=[(1,0,0),(0,1,0),(0,0,1)] {:(implies,x-u=1,y-v=0,z-w=0),(,x+u=0,y+v=1,z+w=0),(,2x+3u=0,2y+3v=0,2z+3w=1):} As this system of equations is inconsistent, we say there is no right inverse for matrix A. The number of right inverses for the matrix [(1,-1,2),(2,-1,1)] is

Let A be an mxxn matrix. If there exists a matrix L of type nxxm such that LA=I_(n) , then L is called left inverse of A. Similarly, if there exists a matrix R of type nxxm such that AR=I_(m) , then R is called right inverse of A. For example, to find right inverse of matrix A=[(1,-1),(1,1),(2,3)] , we take R=[(x,y,x),(u,v,w)] and solve AR=I_(3) , i.e., [(1,-1),(1,1),(2,3)][(x,y,z),(u,v,w)]=[(1,0,0),(0,1,0),(0,0,1)] {:(implies,x-u=1,y-v=0,z-w=0),(,x+u=0,y+v=1,z+w=0),(,2x+3u=0,2y+3v=0,2z+3w=1):} As this system of equations is inconsistent, we say there is no right inverse for matrix A. Which of the following matrices is NOT left inverse of matrix [(1,-1),(1,1),(2,3)]?

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NCERT TAMIL-DETERMINANTS -Miscellaneous Exercises on Chapter 4
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  2. Without expanding the determinant, prove that {:[( a, a ^(2), bc ),( ...

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  3. Evaluate {:[( cos alpha cos beta , cos alpha sin beta , -sin alpha ),...

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  4. If a,b and c are real numbers, and Delta ={:[( b+c,C+a,a+b),( c+a,a...

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  5. Solve the equation {:[( x+a,x,x),(x,x+a,x),(x,x,x+a) ]:}=0,ane 0

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  6. Prove that {:[( a^(2) , bc, ac+c^(2)),( a^(2) +ab,b^(2) ,ac),( ab,b^(2...

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  7. If A^(-1) ={:[( 3,-1,1),(-15,6,-5),(5,-2,2) ]:}and B= {:[( 1,2,-2),( -...

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  8. Let A = {:[( 1,2,1),(2,3,1),(1,1,5) ]:} Verify that (A^(-1) )^(-1) =...

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  9. Evaluate {:[( x,y , x+y),( y,x+y,x),( x+y,x,y)]:}

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  10. Evaluate {:[(1,x,y),( 1,x+y,y),( 1,x,x+y)]:}

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  11. Using properties of determinants in Exercises prove that : {:[( alph...

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  14. {:[( 1,1+p,1+p+q),( 2,3+2p,4+3p+2q),( 3,6+3p,10+6p+3q)]:}=1

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  15. Without expanding evaluate the determinant |(sinalpha,cosalpha,sin(alp...

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  16. Solve system of linear equations , using matrix method 2x+ 3y + ...

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  17. If a,b,c are in A.P. then the determinant {:[( x+2,x+3,x+2a),( x+3,...

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  18. If x,y,z are nonzero real number , then the inverse of matrix A= {:[( ...

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  19. Let A={:[( 1,sin theta , 1),( -sin theta , 1, sin theta ),( -1 ,-sin ...

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