GATE 2024 IN Q.17: solution and answer

Official answer: BSource: Official answer key, GATE 2024 IN (IISc Bengaluru)

1. Poles are the eigenvalues of $A$. They are the roots of the characteristic equation $|sI-A|=0$. For a $2\times 2$ matrix this is $s^2-(\text{tr}A)s+\det A=0$.

2. Condition for a pole at the origin. Put $s=0$ in $|sI-A|=0$: you get $\det (-A)=0$. For a $2\times 2$ matrix $\det (-A)=(-1{)}^2\det A=\det A$, so the condition is $\det A=0$. In other words, $A$ must be singular. (Equivalently, the product of the eigenvalues equals $\det A$, so one eigenvalue is zero exactly when $\det A=0$.)

3. (A) $\det A=(0)(-3)-(1)(-2)=2\ne 0$. The characteristic equation is $s^2+3s+2=0$, so the poles are $-1$ and $-2$. No pole at the origin.

4. (B) $\det A=(1)(3)-(-1.5)(-2)=3-3=0$. This matrix is singular.

5. (C) $\det A=(1)(-3)-(1.5)(2)=-3-3=-6\ne 0$. No pole at the origin.

6. (D) $\det A=(0)(3)-(1)(-2)=2\ne 0$. The characteristic equation is $s^2-3s+2=0$, so the poles are $1$ and $2$. No pole at the origin.

7. Check (B). $\text{tr}A=1+3=4$ and $\det A=0$, so $s^2-4s=s(s-4)=0$. The poles are $s=0$ and $s=4$, so one pole is at the origin. The answer is (B).