ISI2015-MMA-62

Question

If the matrix $A = \begin{bmatrix} a & 1 \\ 2 & 3 \end{bmatrix}$ has $1$ as an eigenvalue, then $\textit{trace}(A)$ is

• A. $4$
• B. $5$
• C. $6$
• D. $7$

Answer 1

Assume other eigen value is $x$.

Sum of the eigen values is equal to the trace of the matrix.

​​​​​​$\implies 1+x=a+3$

Product of eigen values is equal to the determinant of the matrix.

$\implies x=3a-2$

Solving both equations, we get $a=2$

So $trace=a+3=5$

Answer is option (B).