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.

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)

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)

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 – (I–A)3 equals A. (where A ≠ O and I denotes identity matrix) (a) False statement because A−1 exist only if det. A ≠ 0    (b) False statement, as det (AB−1) =det(A).det(B−1)=|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 – (I–A)2 = AB + BA + I – (I + A2 – 2A) = AB + BA + A ∴ True statement   Co-Curriculars

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