The GATE Grind

GATE 2025 EE – Question 14

Engineering Mathematics · Linear Algebra: Matrix Algebra · 1 mark · Multiple choice

Let $P=\begin{bmatrix}2&1&0\\-1&0&0\\0&0&1\end{bmatrix}$ and let $I$ be the identity matrix. Then $P^2$ is equal to

  1. $2P-I$
  2. $P$
  3. $I$
  4. $P+I$

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Correct answer: (A) $2P-I$

Explanation

$P^2=\begin{bmatrix}3&2&0\\-2&-1&0\\0&0&1\end{bmatrix}$ and $2P-I=\begin{bmatrix}3&2&0\\-2&-1&0\\0&0&1\end{bmatrix}$, so $P^2=2P-I$. (Equivalently the characteristic polynomial $(\lambda-1)^3$ gives $(P-I)^2=0$ on the relevant block.)