Which of the following is true for matrix `A=[{:(,1,-1),(,2,3):}]`

Options

A. `A+4I` is a symmetric matrix
B. `A^(2)-4A+5I_(2)=0`
C. A-B is a diagonal matrix for any value of `alpha "if" =[{:(,alpha,-1),(,2,5):}]`
D. A-4I is a skew symmetric matrix.

Answer

To determine which statements are true for the matrix \( A = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix} \), we will analyze each statement step by step.

### Step 1: Calculate \( A + 4I \)

The identity matrix \( I \) of order 2 is given by:
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

Thus, \( 4I \) is:
\[ 4I = 4 \times \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix} \]

Now, we calculate \( A + 4I \):
\[ A + 4I = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix} + \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix} = \begin{pmatrix} 1 + 4 & -1 + 0 \\ 2 + 0 & 3 + 4 \end{pmatrix} = \begin{pmatrix} 5 & -1 \\ 2 & 7 \end{pmatrix} \]

### Step 2: Check if \( A + 4I \) is symmetric

A matrix is symmetric if \( A^T = A \). We find the transpose of \( A + 4I \):
\[ (A + 4I)^T = \begin{pmatrix} 5 & -1 \\ 2 & 7 \end{pmatrix}^T = \begin{pmatrix} 5 & 2 \\ -1 & 7 \end{pmatrix} \]

Since \( (A + 4I)^T \neq A + 4I \), the matrix \( A + 4I \) is not symmetric.

### Step 3: Calculate \( A^2 \)

Now we calculate \( A^2 \):
\[ A^2 = A \cdot A = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix} \cdot \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix} \]

Calculating the elements:
- First row, first column: \( 1 \cdot 1 + (-1) \cdot 2 = 1 - 2 = -1 \)
- First row, second column: \( 1 \cdot (-1) + (-1) \cdot 3 = -1 - 3 = -4 \)
- Second row, first column: \( 2 \cdot 1 + 3 \cdot 2 = 2 + 6 = 8 \)
- Second row, second column: \( 2 \cdot (-1) + 3 \cdot 3 = -2 + 9 = 7 \)

Thus,
\[ A^2 = \begin{pmatrix} -1 & -4 \\ 8 & 7 \end{pmatrix} \]

### Step 4: Calculate \( A^2 - 4A + 5I \)

Now we compute \( 4A \):
\[ 4A = 4 \cdot \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 4 & -4 \\ 8 & 12 \end{pmatrix} \]

And \( 5I \):
\[ 5I = 5 \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} \]

Now we can calculate:
\[ A^2 - 4A + 5I = \begin{pmatrix} -1 & -4 \\ 8 & 7 \end{pmatrix} - \begin{pmatrix} 4 & -4 \\ 8 & 12 \end{pmatrix} + \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} \]

Calculating each element:
- First row, first column: \( -1 - 4 + 5 = 0 \)
- First row, second column: \( -4 + 4 + 0 = 0 \)
- Second row, first column: \( 8 - 8 + 0 = 0 \)
- Second row, second column: \( 7 - 12 + 5 = 0 \)

Thus,
\[ A^2 - 4A + 5I = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

This confirms that \( A^2 - 4A + 5I = 0 \).

### Step 5: Calculate \( A - B \)

Assuming \( B = \begin{pmatrix} \alpha & 2 \\ -1 & 5 \end{pmatrix} \), we compute:
\[ A - B = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix} - \begin{pmatrix} \alpha & 2 \\ -1 & 5 \end{pmatrix} = \begin{pmatrix} 1 - \alpha & -1 - 2 \\ 2 + 1 & 3 - 5 \end{pmatrix} = \begin{pmatrix} 1 - \alpha & -3 \\ 3 & -2 \end{pmatrix} \]

### Step 6: Check if \( A - B \) is a diagonal matrix

A diagonal matrix has non-diagonal elements equal to zero. The matrix \( A - B \) is not a diagonal matrix unless specific values for \( \alpha \) are chosen.

### Step 7: Calculate \( A - 4I \)

Now we calculate \( A - 4I \):
\[ A - 4I = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix} - \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix} = \begin{pmatrix} 1 - 4 & -1 \\ 2 & 3 - 4 \end{pmatrix} = \begin{pmatrix} -3 & -1 \\ 2 & -1 \end{pmatrix} \]

### Step 8: Check if \( A - 4I \) is skew-symmetric

A matrix is skew-symmetric if \( A^T = -A \). We find the transpose:
\[ (A - 4I)^T = \begin{pmatrix} -3 & -1 \\ 2 & -1 \end{pmatrix}^T = \begin{pmatrix} -3 & 2 \\ -1 & -1 \end{pmatrix} \]

Since \( (A - 4I)^T \neq -(A - 4I) \), it is not skew-symmetric.

### Conclusion

The true statements for the matrix \( A \) are:
1. \( A + 4I \) is not symmetric.
2. \( A^2 - 4A + 5I = 0 \) is true.
3. \( A - B \) can be a diagonal matrix for specific values of \( \alpha \).
4. \( A - 4I \) is not skew-symmetric.