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If A is a 3xx3 non-singular matrix such ...

If A is a `3xx3` non-singular matrix such that `A A' = A' A`
and `B = A^(-1)A'` then `B B'` equals to

A

`B^(-1)`

B

`(B^(-1))^T`

C

`I+B`

D

`{:[(2015,0),(1,2015)]:}`

Text Solution

Verified by Experts

The correct Answer is:
D
We have,
`:. B B^T=(A^(-1)A^T)(A^(-1)A^T)^T`
`=(A^(-1)A^T)((A^T)^T(A^(-1))^T)`
`=(A^(-1)A^T)(A(A^(-1))^T) [:'(A^T)^T=A]`
`=A^(-1)(A^TA)(A^(-1))^T`
`=A^(-1)(A A^T)(A^T)^(-1) [:' A^T A=A A^T("given")and(A^(-1))^T=(A^T)^(-1)]`
`=(A^(-1)A)(A^T(A^T)^(-1))=I=I`
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OBJECTIVE RD SHARMA ENGLISH-MATRICES-Chapter Test
  1. If A is a 3xx3 non-singular matrix such that A A' = A' A and B = A^(...

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  2. If A is an invertible matrix and B is a matrix, then

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  3. What is the order of the product [x" "y" "z][{:(a,h,g),(h,b,f),(g,f,c)...

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  4. If {:A=[(a,0,0),(0,b,0),(0,0,c)]:}," then "A^(-1), is

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  5. The inverse of the matrix {:[(1,3),(3,10)]:} is equal to

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  6. If {:A=[(5,2),(3,1)]:}," then "A^(-1)=

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  7. If {:X=[(3,-4),(1,-1)]:}, the value of X^n is equal to

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  8. If {:A=[(5,2),(3,1)]:}," then "A^(-1)=

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  9. For the system of equations: x+2y+3z=1 2x+y+3z=2 5x+5y+9z=4

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  10. If {:A=[(3,1),(-1,2)]:}," then "A^(2)=

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  11. if A=[(4,x+2),(2x-3,x+1)] is symmetric, then x is equal to

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  12. If A+B={:[(1,0),(1,1)]:}andA-2B={:[(-1,1),(0,-1)]:}, then A is equal t...

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  13. {:[(-6,5),(-7,6)]^(-1)=:}

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  14. From the matrix equation AB=AC, we conclude B=C provided.

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  15. If I3 is the identily matrix of order 3, then (I3)^(-1)=

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  16. Let a ,b , c be real numbers. The following system of equations in x ,...

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  17. If A and B are two matrices such that A+B and AB are both defind, then

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  18. A and B are tow square matrices of same order and A' denotes the tran...

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  19. 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)...

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  20. The system of linear equations x+y+z=2,2x+y-z=3, 3x+2y+kz=4 has a uniq...

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  21. If A and B ar square matrices of order 3 such that |A|=-1|B|=3, then |...

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