If `X=[[-x, -y], [z, t]]`, then transpose of adjX is

Options

A. `[[t, z], [-y, -x]]`
B. `[[t, -z], [y, -x]]`
C. `[[t, y], [-z, -x]]`
D. `[[-x, -y], [z, t]]`

Answer

To find the transpose of the adjoint of the matrix \( X = \begin{bmatrix} -x & -y \\ z & t \end{bmatrix} \), we will follow these steps:

### Step 1: Find the Cofactor Matrix

The cofactor matrix is derived from the original matrix \( X \) by calculating the cofactors of each element. The sign pattern for the cofactor matrix is as follows:

\[ \begin{bmatrix} + & - \\ - & + \end{bmatrix} \]

Now, we will calculate the cofactors:

1. **Cofactor of \(-x\)**: The diagonal element is \( t \). Since the sign is positive, the cofactor is \( t \).
2. **Cofactor of \(-y\)**: The diagonal element is \( z \). The sign is negative, so the cofactor is \( -z \).
3. **Cofactor of \( z \)**: The diagonal element is \(-y\). The sign is negative, thus the cofactor is \( -(-y) = y \).
4. **Cofactor of \( t \)**: The diagonal element is \(-x\). The sign is positive, so the cofactor is \( -x \).

Thus, the cofactor matrix is:

\[ \text{Cofactor Matrix} = \begin{bmatrix} t & -z \\ y & -x \end{bmatrix} \]

### Step 2: Find the Adjoint Matrix

The adjoint of a matrix is the transpose of its cofactor matrix. Therefore, we will transpose the cofactor matrix:

\[ \text{Adjoint of } X = \begin{bmatrix} t & -z \\ y & -x \end{bmatrix}^T = \begin{bmatrix} t & y \\ -z & -x \end{bmatrix} \]

### Step 3: Find the Transpose of the Adjoint Matrix

Now we need to find the transpose of the adjoint matrix:

\[ \text{Transpose of Adjoint of } X = \begin{bmatrix} t & y \\ -z & -x \end{bmatrix}^T = \begin{bmatrix} t & -z \\ y & -x \end{bmatrix} \]

### Final Result

Thus, the transpose of the adjoint of matrix \( X \) is:

\[ \text{Transpose of } \text{adjoint } X = \begin{bmatrix} t & -z \\ y & -x \end{bmatrix} \]