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GO Classes DPP | GATE CS, DA | Linear Algebra | Cayley-Hamilton Theorem

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5 5 votes

Let \(N\) be the matrix
\[
N=\begin{bmatrix}1&2\\3&4\end{bmatrix}.
\]
Which of the following matrix equations does \(N\) satisfy?

  1. \(N^{2}-5N-2I=0\)
     
  2. \(N^{2}+5N-2I=0\)
     
  3. \(N^{2}-5N+2I=0\)
     
  4. \(N^{2}+5N+2I=0\)

4 Answers

2 2 votes

Let $N=\begin{bmatrix}1&2\\3&4\end{bmatrix}$.

First compute trace and determinant:

$\text{trace}(N)=1+4=5$

$\det(N)=1\cdot 4-2\cdot 3=4-6=-2$

For a $2\times 2$ matrix, the characteristic polynomial is

$p(\lambda)=\lambda^2-(\text{trace}(N))\lambda+\det(N)$

So,

$p(\lambda)=\lambda^2-5\lambda-2$

By the Cayley-Hamilton theorem, $p(N)=0$, hence

$N^2-5N-2I=0$

Therefore, the correct option is  $\boxed{\text{A}}$

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