Let A=[array|c|c| 1 & 2 1 & array] and B=[array|c|c| 3 & 3 & 2 array].

Question

Let $A=[\begin{array}{|c|c|} 1 & 2 \ 1 & \alpha \end{array}]$ and $B=[\begin{array}{|c|c|} 3 & 3 \ \beta  & 2 \end{array}]$. If ${A}^{2}-4A+I=O$ and ${B}^{2}-5B-6I=O$, then among the two statements:

(S1) : $[(B-A)(B+A){]}^{T}=[\begin{array}{|c|c|} 13 & 15 \ 7 & 10 \end{array}]$

and

(S2) : $det(adj(A+B))=-5$

Accepted Answer

$\begin{gathered} A=[1 2 \ 1 \alpha ] \ {A}^{2}-4A+I=0 $

trace of matrix

$ \alpha +1=4 \ \alpha =3 \ B=[3 3 \ \beta 2] \ {B}^{2}-5B-6I=0 $

$\begin{gathered} \det .(B)=6-3\beta =-6 \ 3\beta =12 \ \beta =4 \ B-A=[2 \quad 1 \ 3 \quad -1];B+A=[4 \quad 5 \ 5 \quad 5] \ (B-A)(B+A)=[13 \quad 15 \ 7 \quad 10] \ adj(A)=[3 \quad -2 \ -1 \quad 1],adjB=[2 \quad -3 \ -4 \quad 3] \ adj(A+B)=[5 \quad -5 \ -5 \quad 4 \end{gathered}] \end{gathered}$

Solutions · 1