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 )|A|=1 the trace of matrix A is 0.  (the trace of a square matrix is the sum of elements on the main diagonal)∣∣A−|A|adj(A)∣∣=2     ∣∣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|=a1a4−a2a3=k(say) adj(A)=[a4−a2−a3a1] ∣∣A+|A|adj(A)∣∣=0 ⇒[a1+ka4a2−ka2a3−ka3a4+ka1]=0 ⇒k[1+a24+a21+k2+2a2a3]=0 ⇒(a4+a1)2+1+k2−2(a1a4−a2a3)=0⇒(a4+a1)2+1+k2−2k=0 ⇒k=|A|=1  &  a4+a1=0 ∣∣A−|A|adj(A)∣∣=∣∣A−adj(A)∣∣  (∵|A|=1) =[a1−a42a22a3a4−a1] =2a1a4−a21−a24−4a2a3 =4a1a4−(2a1a4+a21+a24)−4a2a3 =4(a1a4−a2a3)−(a1+a4)2 =4−0 =4

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