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If nth-order square matrix A is a orthog...

If nth-order square matrix A is a orthogonal, then `|"adj (adj A)"|` is

A

always -1 if n is even

B

always 1 if n is odd

C

always 1

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Since A is orthogonal, we have
`A A^(T)=I`
or `|A A^(T)|=1`
or `|A^(2)|=1`
or `|A|= pm 1`
Now `|"adj. (adj. A)"|=|A|^((n-1)^(2))`
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CENGAGE-MATRICES-Exercise (Single)
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  2. If A=[(0,1,2),(1,2,3),(3,a,1)] and A^(-1)=[(1//2,-1//2,1//2),(-4,3,c),...

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  3. If nth-order square matrix A is a orthogonal, then |"adj (adj A)"| is

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  4. Let aa n db be two real numbers such that a >1,b > 1. If A=(a0 0b) , t...

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  5. If A=[a("ij")](4xx4), such that a("ij")={(2",","when "i=j),(0",","when...

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  6. A is an involuntary matrix given by A=[0 1-1 4-3 4 3-3 4] , then the i...

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  7. If A is a nonsingular matrix such that A A^(T)=A^(T)A and B=A^(-1) A^(...

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  8. If P is an orthogonal matrix and Q=P A P^T an dx=P^T A b. I c. A^(100...

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  9. If Aa n dB are two non-singular matrices of the same order such that B...

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  10. If adjB=A ,|P|=|Q|=1,then adj(Q^(-1)B P^(-1)) is

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  11. If A is non-singular and (A-2I)(A-4I)=O ,t h e n1/6A+4/3A^(-1) is equa...

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  12. Let f(x)=(1+x)/(1-x) . If A is matrix for which A^3=O ,t h e nf(A) is ...

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  13. if[{:(2,1),(3,2):}]A[{:(-3,2),(5,-3):}]=[{:(1,0),(0,1):}],"then" A=?

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  14. If A^2-A +I = 0, then the inverse of A is: (A) A+I (B) A (C) ...

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  15. If F(x)=[("cos"x,-sin x,0),(sin x,cos x,0),(0,0,1)] and G(y)=[(cos y,0...

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  16. If Aa n dB are square matrices of the same order and A is non-singular...

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  17. If k in Rot h e ndet{a d j(k In)} is equal to K^(n-1) b. K^(n(n-1)) c...

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  18. Given that matrix A[(x,3,2),(1,y,4),(2,2,z)]. If xyz=60 and 8x+4y+3z=2...

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  19. Let A=[(1,2,3),(2,0,5),(0,2,1)] and B=[(0),(-3),(1)]. Which of the fol...

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  20. If A is a square matrix of order less than 4 such that |A-A^(T)| ne 0 ...

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