Question

Suppose matrix $A=\left[\begin{array}{cc}2& T\\ -1& 6\end{array}\right]$ and X is a 2 $\times$ 1 non-null column vector, then AX = $\lambda$IX for what value(s) of $\lambda$? (I is the identity matrix of order 2)

• A. 5
• B. -5
• C. 1
• D. -1

Answer 1

Answer: (B), (C)

The correct options are
B

-5

C

1

AX = $\lambda$IX $\Rightarrow$ (A - $\lambda$I)X = 0
Since X is not a null vector, $|A-\lambda I|=0$
$A-\lambda I=\left[\begin{array}{cc}2& 7\\ -1& 6\end{array}\right]-\left[\begin{array}{cc}\lambda & 0\\ 0& \lambda \end{array}\right]=\left[\begin{array}{cc}2-\lambda & 7\\ -1& 6-\lambda \end{array}\right]$
$|A-\lambda I|=(2-\lambda )(6-\lambda )+7={\lambda }^2+4\lambda -5=(\lambda +5)(\lambda -1)|A-\lambda I|=0\Rightarrow \lambda =-5or\lambda =1$
(b) and (c) are the correct options.