To find the conjugate transpose (also known as the Hermitian transpose) of a matrix, you need to follow these steps:
1. Transpose the matrix: Swap rows with columns.
2. Conjugate each element: Replace each element with its complex conjugate.
Given matrix:
[tex]\[
\begin{pmatrix}
1 & 2+i \\
5 & 5 \\
9+7i & -3-3i
\end{pmatrix}
\][/tex]
### Step 1: Transpose the Matrix
The transpose of the matrix is obtained by swapping rows with columns:
[tex]\[
\begin{pmatrix}
1 & 5 & 9+7i \\
2+i & 5 & -3-3i
\end{pmatrix}
\][/tex]
### Step 2: Conjugate Each Element
To take the complex conjugate of each element:
- For a real number [tex]\(a\)[/tex], the complex conjugate is the number itself, [tex]\(a\)[/tex].
- For a complex number [tex]\(a + bi\)[/tex], the complex conjugate is [tex]\(a - bi\)[/tex].
Applying this to the transposed matrix:
[tex]\[
\begin{pmatrix}
1 & 5 & 9+7i \\
2+i & 5 & -3-3i
\end{pmatrix}
\][/tex]
1. [tex]\(1 \rightarrow 1 - 0i = 1\)[/tex]
2. [tex]\(5 \rightarrow 5 - 0i = 5\)[/tex]
3. [tex]\(9+7i \rightarrow 9-7i\)[/tex]
4. [tex]\(2+i \rightarrow 2-i\)[/tex]
5. [tex]\(5 \rightarrow 5 - 0i = 5\)[/tex]
6. [tex]\(-3-3i \rightarrow -3+3i\)[/tex]
Thus, the conjugated elements are:
[tex]\[
\begin{pmatrix}
1 & 5 & 9-7i \\
2-i & 5 & -3+3i
\end{pmatrix}
\][/tex]
### Final Answer:
The conjugate transpose of the given matrix is:
[tex]\[
\begin{pmatrix}
1 & 5 & 9-7i \\
2-i & 5 & -3+3i
\end{pmatrix}
\][/tex]
