Let $P=\left[\begin{array}{ccc}2& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]$ and let $I$ be the identity matrix. Then $P^2$ is equal to

The question asks us to compute the square of the matrix $P$, denoted as $P^2$, and express it in the form $aP+bI$ or similar, where $I$ is the identity matrix.

**Calculate $P^2$ using Matrix Multiplication**

Given the matrix $P=\left[\begin{array}{ccc}2& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]$, we calculate $P^2$ by multiplying $P$ by itself:

$P^2=P\times P=\left[\begin{array}{ccc}2& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]\times \left[\begin{array}{ccc}2& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]$

Performing the matrix multiplication:

- Element (1,1): $(2)(2)+(1)(-1)+(0)(0)=4-1+0=3$
- Element (1,2): $(2)(1)+(1)(0)+(0)(0)=2+0+0=2$
- Element (1,3): $(2)(0)+(1)(0)+(0)(1)=0+0+0=0$
- Element (2,1): $(-1)(2)+(0)(-1)+(0)(0)=-2+0+0=-2$
- Element (2,2): $(-1)(1)+(0)(0)+(0)(0)=-1+0+0=-1$
- Element (2,3): $(-1)(0)+(0)(0)+(0)(1)=0+0+0=0$
- Element (3,1): $(0)(2)+(0)(-1)+(1)(0)=0+0+0=0$
- Element (3,2): $(0)(1)+(0)(0)+(1)(0)=0+0+0=0$
- Element (3,3): $(0)(0)+(0)(0)+(1)(1)=0+0+1=1$

Therefore, $P^2=\left[\begin{array}{ccc}3& 2& 0\\ -2& -1& 0\\ 0& 0& 1\end{array}\right]$.

**Evaluate Options in Terms of $P$ and $I$ **

Now, we check which of the given options equals the calculated $P^2$. Let $I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$.

**Option 1: $2P-I$ **

First, calculate $2P$:

$2P=2\times \left[\begin{array}{ccc}2& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}4& 2& 0\\ -2& 0& 0\\ 0& 0& 2\end{array}\right]$

Now, calculate $2P-I$:

$2P-I=\left[\begin{array}{ccc}4& 2& 0\\ -2& 0& 0\\ 0& 0& 2\end{array}\right]-\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}4-1& 2-0& 0-0\\ -2-0& 0-1& 0-0\\ 0-0& 0-0& 2-1\end{array}\right]=\left[\begin{array}{ccc}3& 2& 0\\ -2& -1& 0\\ 0& 0& 1\end{array}\right]$

This matches the calculated $P^2$.

**Option 2: $P$ **

$P=\left[\begin{array}{ccc}2& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]$. This does not equal $P^2$.

**Option 3: $I$ **

$I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$. This does not equal $P^2$.

**Option 4: $P+I$ **

$P+I=\left[\begin{array}{ccc}2& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}3& 1& 0\\ -1& 1& 0\\ 0& 0& 2\end{array}\right]$. This does not equal $P^2$.

Conclusion

The calculation shows that $P^2$ is equal to $2P-I$. This corresponds to Option A.