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Question

Which of the following statement(s) is (are) correct ?



A

If A, B and C are square matrices of order 3 such that AB = AC and det(A) = 0, then B = C.

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B

If A = diag(2, 1, – 3) and B = diag(1, 1, 2), then det(AB1) = 3.
(where diag (a, b, c) denotes diagonal matrix)

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C

If A=111111111, then A3=9A

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D

If A is square matrix of order 3 such that A2 = A and B = I – A, then AB + BA + I – (IA)3 equals A. (where A O and I denotes identity matrix)

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Solution

The correct options are
C

If A=111111111, then A3=9A


D

If A is square matrix of order 3 such that A2 = A and B = I – A, then AB + BA + I – (IA)3 equals A. (where A O and I denotes identity matrix)


(a) False statement because A1 exist only if det. A 0    (b) False statement, as det (AB1)
=det(A).det(B1)=|A||B|=62=3
(c) True statement,       A2=333333333=3A
  A3=3A2=3(3A)A3=9A
(d) Given A2 = A and B = I – A
Now AB + BA + I – (IA)2
= AB + BA + I – (I + A2 – 2A) = AB + BA + A
True statement
 


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