To verify whether the matrices [tex]\( P^T \)[/tex] and [tex]\( P^T P \)[/tex] are symmetric, follow these steps:
1. Define Matrix [tex]\( P \)[/tex]:
We have [tex]\( P \)[/tex] as:
[tex]\[
P = \begin{pmatrix}
3 \\
-4 \\
5
\end{pmatrix}
\][/tex]
2. Compute the Transpose [tex]\( P^T \)[/tex]:
The transpose of a matrix is obtained by switching rows and columns. For the given column vector [tex]\( P \)[/tex]:
[tex]\[
P^T = \begin{pmatrix}
3 & -4 & 5
\end{pmatrix}
\][/tex]
3. Check if [tex]\( P^T \)[/tex] is Symmetric:
A matrix [tex]\( A \)[/tex] is symmetric if [tex]\( A = A^T \)[/tex]. Compute the transpose of [tex]\( P^T \)[/tex]:
[tex]\[
(P^T)^T = \begin{pmatrix}
3 \\
-4 \\
5
\end{pmatrix}
\][/tex]
Observe that [tex]\((P^T)^T = P\)[/tex], but [tex]\( P \neq P^T \)[/tex]. Hence, [tex]\( P^T \)[/tex] is not symmetric.
4. Compute the Product [tex]\( P^T P \)[/tex]:
[tex]\[
P^T P = \begin{pmatrix}
3 & -4 & 5
\end{pmatrix}
\begin{pmatrix}
3 \\
-4 \\
5
\end{pmatrix}
\][/tex]
Calculate each element of the resulting matrix:
[tex]\[
P^T P = (3 \cdot 3) + (-4 \cdot -4) + (5 \cdot 5) = 9 + 16 + 25 = 50
\][/tex]
Thus,
[tex]\[
P^T P = \begin{pmatrix}
50
\end{pmatrix}
\][/tex]
Since the result is a [tex]\( 1 \times 1 \)[/tex] matrix, it is inherently symmetric.
5. Conclusion:
- The matrix [tex]\( P^T \)[/tex] is not symmetric.
- The matrix [tex]\( P^T P \)[/tex] is symmetric.
Thus, the verifications yield that:
- [tex]\( P^T \)[/tex] is not symmetric.
- [tex]\( P^T P \)[/tex] is symmetric.
