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

If A is an `3xx3` non-singular matrix such that `A A^T=A^TA and B=A^(-1)A^T," then " B B^T` equals

A

`B^(-1)`

B

`(B^(-1))`

C

`I+B`

D

`I`

Text Solution

Verified by Experts

The correct Answer is:
D
`B = A^(-1) A'`
`B' = (A ^(-1) A')' = A(A^(-1))'`
Now, `BB' = (A^(-1)A') A(A^(-1)) ' = A^(-1) (A'A) (A^(-1))'`
`= A^(-1) (A A')(A^(-1)) " " [because A' A = A A' ]`
`= (A^(-1) A) A'(A^(-1))'`
`= (IA') (A^(-1))'=A^'(A^(-1))' = (A^(-1)A) ' = I' = I`
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ARIHANT MATHS-MATRICES -Exercise (Questions Asked In Previous 13 Years Exam)
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  2. Let A and B two symmetric matrices of order 3. Statement 1 : A(BA) a...

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  3. Let P=[a(i j)] be a 3xx3 matrix and let Q=[b(i j)],w h e r eb(i j)=2^(...

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  4. If P is a 3xx3 matrix such that P^T = 2P+I, where P^T is the transpose...

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  5. If the adjoint of a 3x3 matrix P is (1 4 4) (2 1 7) (1 1 3) , t...

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  6. Let A=((1,0,0),(2,1,0),(3,2,1)). If u(1) and u(2) are column matrices ...

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  7. Let P and Q be 3xx3 matrices P ne Q. If P^(3)=Q^(3) and P^(2)Q=Q^(2)P,...

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  8. IF P=[(1,alpha,3),(1,3,3),(2,4,4)] is the adjoint of 3xx3 matrix A and...

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  9. For 3xx3 matrices M \ a n d \ N , which of the following statement (s)...

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  10. Let omega be a complex cube root of unity with omega!=1a n dP=[p(i j)]...

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  11. If A is an 3xx3 non-singular matrix such that A A^T=A^TA and B=A^(-1)A...

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  12. Let M be a 2xx2 symmetric matrix with integer entries. Then , M is i...

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  13. Let M and N be two 3xx3 matrices such that MN=NM. Further, if M ne N^(...

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  14. If A=[(1,2,2),(2,1,-2),(a,2,b)] is a matrix satisying the equation A A...

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  15. Let X \ a n d \ Y be two arbitrary, 3xx3 , non-zero, skew-symmetric ma...

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  16. If A=[(5a,-b),(3,2)] and A adj A=A A^(T), then 5a+b is equal to

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  17. Let p=[(3,-1,-2),(2,0,alpha),(3,-5,0)], where alpha in RR. Suppose Q=[...

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  18. Let z=(-1+sqrt(3)i)/(2), where i=sqrt(-1), and r, s in {1, 2, 3}. Let ...

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  19. Let P=[(1,0,0),(3,1,0),(9,3,1)] and Q = [q(ij)] be two 3xx3 matrices s...

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  20. If A=[(2,-3),(-4,1)], then adj (3A^(2)+12 A) is equal to

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