GATE 2024 EE Q.32: solution and answer

Official answer: 29Source: Official answer key, GATE 2024 EE (IISc Bengaluru)

1. Let $B=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, so $A=B^2$. For any square matrix, the sum of the eigenvalues equals the trace: $\sum {\lambda }_i=\mathrm{tr}(A)$.

2. Square the matrix: $$A=B^2=\left[\begin{array}{cc}1\cdot 1+2\cdot 3& 1\cdot 2+2\cdot 4\\ 3\cdot 1+4\cdot 3& 3\cdot 2+4\cdot 4\end{array}\right]=\left[\begin{array}{cc}7& 10\\ 15& 22\end{array}\right]$$

3. Trace: $\mathrm{tr}(A)=7+22=29$.

4. Check using the eigenvalues of $B$. If ${\lambda }_1,{\lambda }_2$ are the eigenvalues of $B$, the eigenvalues of $B^2$ are ${\lambda }_1^2,{\lambda }_2^2$. For $B$, ${\lambda }_1+{\lambda }_2=\mathrm{tr}(B)=5$ and ${\lambda }_1{\lambda }_2=\det B=4-6=-2$.

5. So ${\lambda }_1^2+{\lambda }_2^2=({\lambda }_1+{\lambda }_2{)}^2-2{\lambda }_1{\lambda }_2=25-2(-2)=29$, which agrees with the trace. (Explicitly, ${\lambda }_{1,2}=\frac{5\pm \sqrt{33}}{2}$, and their squares add to $\frac{2(25+33)}{4}=29$.)

6. The sum of the eigenvalues of $A$ is $29$. It is already an integer, so no rounding is needed.