If `A=[[0, 1, -2], [-1, 0, 5], [2, -5, 0]]`, then

Options

A. `A'=-A`
B. `A'=A`
C. `A'=2A`
D. `A'=-2A`

Answer

To determine the properties of the matrix \( A = \begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & 5 \\ 2 & -5 & 0 \end{bmatrix} \), we will check if it is a skew-symmetric matrix and verify its transpose.

### Step 1: Check if the matrix is skew-symmetric
A matrix \( A \) is skew-symmetric if \( A^T = -A \) and all diagonal elements are zero.

**Diagonal Elements:**
- The diagonal elements of \( A \) are \( 0, 0, 0 \) (which are all zero).

**Off-diagonal Elements:**
- For the matrix to be skew-symmetric, the following conditions must hold:
- \( A_{12} = -A_{21} \) (i.e., \( 1 = -(-1) \))
- \( A_{13} = -A_{31} \) (i.e., \( -2 = -2 \))
- \( A_{23} = -A_{32} \) (i.e., \( 5 = -(-5) \))

Since all these conditions are satisfied, \( A \) is indeed a skew-symmetric matrix.

### Step 2: Calculate the transpose of matrix \( A \)
To find the transpose \( A^T \), we swap the rows and columns of \( A \):

\[ A^T = \begin{bmatrix} 0 & -1 & 2 \\ 1 & 0 & -5 \\ -2 & 5 & 0 \end{bmatrix} \]

### Step 3: Calculate \(-A\)
Now, we calculate \(-A\):

\[ -A = -\begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & 5 \\ 2 & -5 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -1 & 2 \\ 1 & 0 & -5 \\ -2 & 5 & 0 \end{bmatrix} \]

### Step 4: Compare \( A^T \) and \(-A\)
Now we compare \( A^T \) and \(-A\):

\[ A^T = \begin{bmatrix} 0 & -1 & 2 \\ 1 & 0 & -5 \\ -2 & 5 & 0 \end{bmatrix} \]
\[ -A = \begin{bmatrix} 0 & -1 & 2 \\ 1 & 0 & -5 \\ -2 & 5 & 0 \end{bmatrix} \]

Since \( A^T = -A \), we conclude that \( A \) is skew-symmetric.

### Final Conclusion
The matrix \( A \) is a skew-symmetric matrix, and we have verified that \( A^T = -A \).

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