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Question

If A be a non-singular matrix of order 2, such that A+|A|adj(A)=0, then which of the following option(s) is/are always correct ?
(where adj(A) is the adjoint of matrix A )
  1. |A|=1 
  2. the trace of matrix A is 0. 
    (the trace of a square matrix is the sum of elements on the main diagonal)
  3. A|A|adj(A)=2     
  4. A|A|adj(A)=4      


Solution

The correct options are
A |A|=1 
B the trace of matrix A is 0. 
(the trace of a square matrix is the sum of elements on the main diagonal)
D A|A|adj(A)=4      
Let A=[a1a2a3a4]
|A|=a1a4a2a3=k(say)

adj(A)=[a4a2a3a1]

A+|A|adj(A)=0
[a1+ka4a2ka2a3ka3a4+ka1]=0
k[1+a24+a21+k2+2a2a3]=0
(a4+a1)2+1+k22(a1a4a2a3)=0(a4+a1)2+1+k22k=0
k=|A|=1  &  a4+a1=0

A|A|adj(A)=Aadj(A)  (|A|=1)
=[a1a42a22a3a4a1]
=2a1a4a21a244a2a3
=4a1a4(2a1a4+a21+a24)4a2a3
=4(a1a4a2a3)(a1+a4)2
=40
=4

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