GATE 2023 BT – Question 58
If $$A=\begin{pmatrix}1&2\\3&5\end{pmatrix},$$ the value of $|A^4+3A^2-5A+6I|$ is ___________.
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Correct answer: 10551
Explanation
**Step 1: characteristic equation** of $A=\begin{pmatrix}1&2\\3&5\end{pmatrix}$: trace $=6$, determinant $=5-6=-1$, so
$$\lambda^2-6\lambda-1=0\;\Rightarrow\;A^2=6A+I \quad(\text{Cayley-Hamilton}).$$
**Step 2: higher powers.**
$$A^4=(6A+I)^2=36A^2+12A+I=36(6A+I)+12A+I=228A+37I .$$
**Step 3: the polynomial.**
$$A^4+3A^2-5A+6I=(228A+37I)+3(6A+I)-5A+6I=241A+46I .$$
**Step 4: its determinant.** For $aA+bI$ with $A=\begin{pmatrix}1&2\\3&5\end{pmatrix}$:
$$\det=(a+b)(5a+b)-6a^2=-a^2+6ab+b^2 .$$
With $a=241,\ b=46$:
$$-58081+66516+2116=\mathbf{10551}.$$