Let's approach this problem step-by-step.
Step 1: Understanding the Problem
We need to show that two given matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are equal to their transposes. Mathematically, this means proving that:
1. [tex]\(A^T = A\)[/tex]
2. [tex]\(B^T = B\)[/tex]
Matrices that are equal to their own transposes are called "symmetric."
Step 2: Write Down the Matrices
The given matrices are:
[tex]\[ A = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix} \][/tex]
[tex]\[ B = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix} \][/tex]
Step 3: Calculate the Transpose of Matrix [tex]\(A\)[/tex]
The transpose of a matrix is found by swapping its rows with its columns.
For matrix [tex]\(A\)[/tex]:
- The first row [tex]\((1, 0)\)[/tex] becomes the first column:
[tex]\[
\begin{pmatrix}
1 \\
0
\end{pmatrix}
\][/tex]
- The second row [tex]\((0, 1)\)[/tex] becomes the second column:
[tex]\[
\begin{pmatrix}
0 \\
1
\end{pmatrix}
\][/tex]
Thus, the transpose of [tex]\(A\)[/tex] (denoted as [tex]\(A^T\)[/tex]) is:
[tex]\[ A^T = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix} \][/tex]
Comparing [tex]\(A^T\)[/tex] with [tex]\(A\)[/tex], we see that [tex]\(A^T = A\)[/tex].
Step 4: Calculate the Transpose of Matrix [tex]\(B\)[/tex]
Similarly, for matrix [tex]\(B\)[/tex]:
- The first row [tex]\((1, 0, 0)\)[/tex] becomes the first column:
[tex]\[
\begin{pmatrix}
1 \\
0 \\
0
\end{pmatrix}
\][/tex]
- The second row [tex]\((0, 1, 0)\)[/tex] becomes the second column:
[tex]\[
\begin{pmatrix}
0 \\
1 \\
0
\end{pmatrix}
\][/tex]
- The third row [tex]\((0, 0, 1)\)[/tex] becomes the third column:
[tex]\[
\begin{pmatrix}
0 \\
0 \\
1
\end{pmatrix}
\][/tex]
Thus, the transpose of [tex]\(B\)[/tex] (denoted as [tex]\(B^T\)[/tex]) is:
[tex]\[ B^T = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix} \][/tex]
Comparing [tex]\(B^T\)[/tex] with [tex]\(B\)[/tex], we see that [tex]\(B^T = B\)[/tex].
Step 5: Summarize Your Findings
We have proven that:
a) [tex]\(A^T = A\)[/tex]
b) [tex]\(B^T = B\)[/tex]
This means that both matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are symmetric.
In a sentence: Both the given matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are symmetric, as they are equal to their respective transposes.
