If `A= [[1,-1,-1],[1,-1,0],[1,0,-1]]` then find transpose of A

Explanation

To find the transpose of the matrix \( A \), we will follow these steps:

Given matrix:
\[ A = \begin{bmatrix} 1 & -1 & -1 \\ 1 & -1 & 0 \\ 1 & 0 & -1 \end{bmatrix} \]

### Step 1: Identify the dimensions of matrix A
Matrix \( A \) has 3 rows and 3 columns.

**Hint:** The transpose of a matrix switches its rows and columns.

### Step 2: Write down the elements of the transpose
To find the transpose \( A^T \), we will convert the rows of \( A \) into columns.

- The first row of \( A \) is \( [1, -1, -1] \), which will become the first column of \( A^T \).
- The second row of \( A \) is \( [1, -1, 0] \), which will become the second column of \( A^T \).
- The third row of \( A \) is \( [1, 0, -1] \), which will become the third column of \( A^T \).

### Step 3: Construct the transpose matrix
Now we can construct the transpose matrix \( A^T \):
\[ A^T = \begin{bmatrix} 1 & 1 & 1 \\ -1 & -1 & 0 \\ -1 & 0 & -1 \end{bmatrix} \]

### Final Result
Thus, the transpose of matrix \( A \) is:
\[ A^T = \begin{bmatrix} 1 & 1 & 1 \\ -1 & -1 & 0 \\ -1 & 0 & -1 \end{bmatrix} \]

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