Question

Consider the following 2 x 2 matrix A where two elements are unknown and are marked by a and b. The eigen values of this matrix are - 1 and 7. What are the values of a and b? $A=\left[\begin{array}{cc}1& 4\\ b& a\end{array}\right]$

• A. $\begin{array}{cc}a=6,& b=4\end{array}$
• B. $\begin{array}{cc}a=4,& b=6\end{array}$
• C. $\begin{array}{cc}a=3,& b=5\end{array}$
• D. $\begin{array}{cc}a=5,& b=3\end{array}$

Answer 1

The correct option is D $\begin{array}{cc}a=5,& b=3\end{array}$

Given that $A=\left[\begin{array}{cc}1& 4\\ b& a\end{array}\right]$ has given values -1 and 7.

So Trace (A) $={\lambda }_1+{\lambda }_{2}and|A|={\lambda }_1.{\lambda }_2$
$a+1=-1+7$
and $a-4b=(-1)\left(7\right)$
$a=5$
and $b=3$