Question

The eigen values of a 2 x 2 matrix X are -2 and -3. The eigen values of matrix $(X+I{)}^{-1}(X+5I)$ are

• A. $-3,-4$
• B. $-1,-2$
• C. $-1,-3$
• D. $-2,-4$

Answer 1

The correct option is C $-1,-3$

Eigen values of X are $-2,-3$
Eigen values of I are 1, 1
Eigen values of $X+I$ are $(-2+1)$&$(-3+2)$ i.e, $-1,-2$
$\Rightarrow$ Eigen values of $(X+I{)}^{-1}$ are $-1,-\frac{1}{2}$
Given $(X+I{)}^{-1}(X+5I)=(X+I{)}^{-1}(X+I+4I)$
$=(X+1{)}^{-1}(X+1)+4(X+1{)}^{-1}$
$=I+4(X+I{)}^{-1}$
$\therefore$ The eigen values of
$I+4(X+I{)}^{-1}=1+4(-1)$&$1+4(-\frac{1}{2})$
$=-3$ & $-1$