Find the transpose of each of the following matrices:(i) `[(5 , 1), (2 ,1) ]` (ii) `[(1 , -1), (2 ,3) ]` (iii) `[(-1 , 5, 6), (sqrt 3 ,5, 6), (2 , 3, -1)]`

Explanation

To find the transpose of the given matrices, we will follow the definition of the transpose operation, which involves converting the rows of the matrix into columns and the columns into rows.

### Step-by-Step Solution:

**(i)** For the matrix \( A = \begin{pmatrix} 5 & 1 \\ 2 & 1 \end{pmatrix} \):

1. Identify the rows of matrix \( A \):
- Row 1: \( (5, 1) \)
- Row 2: \( (2, 1) \)

2. Convert these rows into columns for the transpose:
- Column 1: \( (5, 2) \)
- Column 2: \( (1, 1) \)

3. Thus, the transpose of matrix \( A \) is:
\[ A^T = \begin{pmatrix} 5 & 2 \\ 1 & 1 \end{pmatrix} \]

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**(ii)** For the matrix \( B = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix} \):

1. Identify the rows of matrix \( B \):
- Row 1: \( (1, -1) \)
- Row 2: \( (2, 3) \)

2. Convert these rows into columns for the transpose:
- Column 1: \( (1, 2) \)
- Column 2: \( (-1, 3) \)

3. Thus, the transpose of matrix \( B \) is:
\[ B^T = \begin{pmatrix} 1 & 2 \\ -1 & 3 \end{pmatrix} \]

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**(iii)** For the matrix \( C = \begin{pmatrix} -1 & 5 & 6 \\ \sqrt{3} & 5 & 6 \\ 2 & 3 & -1 \end{pmatrix} \):

1. Identify the rows of matrix \( C \):
- Row 1: \( (-1, 5, 6) \)
- Row 2: \( (\sqrt{3}, 5, 6) \)
- Row 3: \( (2, 3, -1) \)

2. Convert these rows into columns for the transpose:
- Column 1: \( (-1, \sqrt{3}, 2) \)
- Column 2: \( (5, 5, 3) \)
- Column 3: \( (6, 6, -1) \)

3. Thus, the transpose of matrix \( C \) is:
\[ C^T = \begin{pmatrix} -1 & \sqrt{3} & 2 \\ 5 & 5 & 3 \\ 6 & 6 & -1 \end{pmatrix} \]

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### Summary of Transposes:
- \( A^T = \begin{pmatrix} 5 & 2 \\ 1 & 1 \end{pmatrix} \)
- \( B^T = \begin{pmatrix} 1 & 2 \\ -1 & 3 \end{pmatrix} \)
- \( C^T = \begin{pmatrix} -1 & \sqrt{3} & 2 \\ 5 & 5 & 3 \\ 6 & 6 & -1 \end{pmatrix} \)

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