Given P = \(\begin{bmatrix}1 & 2\\2 & 3\end{bmatrix}\), find P\(^2\) – 4P – I where I is the identity matrix
- \(\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}\)
-
\(\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}\)
- \(\begin{bmatrix}-1 & 0\\0 & -1\end{bmatrix}\)
- \(\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\)
The correct answer is: B
Explanation
Given the matrix
\[
P = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix},
\]
we want to find \( P^2 - 4P - I \), where \( I \) is the identity matrix:
\[
I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.
\]
Calculate {\( P^2 \)}
\[
P^2 = P \times P = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 5 & 8 \\ 8 & 13 \end{bmatrix}.
\]
Calculate {\( 4P \)}
\[
4P = 4 \times \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 4 & 8 \\ 8 & 12 \end{bmatrix}.
\]
Calculate {\( P^2 - 4P \)}
\[
P^2 - 4P = \begin{bmatrix} 5 & 8 \\ 8 & 13 \end{bmatrix} - \begin{bmatrix} 4 & 8 \\ 8 & 12 \end{bmatrix} = \begin{bmatrix} 5 - 4 & 8 - 8 \\ 8 - 8 & 13 - 12 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.
\]
Calculate {\( P^2 - 4P - I \)}
\[
P^2 - 4P - I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}.
\]
Thus, the result of \( P^2 - 4P - I \) is
\[
\boxed{\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}}.
\]
There is an explanation video available .