If `{:A=[(alpha,0),(1,1)]andB=[(1,0),(5,1)]:}`, then the value of `alpha` for which `A^2=B`, is

Options

A. 1
B. -1
C. 4
D. no real values

Explanation

To find the value of \( \alpha \) for which \( A^2 = B \), we will first calculate \( A^2 \) and then set it equal to matrix \( B \).

Given:
\[ A = \begin{pmatrix} \alpha & 0 \\ 1 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 5 & 1 \end{pmatrix} \]

### Step 1: Calculate \( A^2 \)
To calculate \( A^2 \), we perform the matrix multiplication \( A \times A \):
\[ A^2 = A \times A = \begin{pmatrix} \alpha & 0 \\ 1 & 1 \end{pmatrix} \times \begin{pmatrix} \alpha & 0 \\ 1 & 1 \end{pmatrix} \]

### Step 2: Perform the multiplication
The resulting matrix \( A^2 \) will be:
\[ A^2 = \begin{pmatrix} \alpha \cdot \alpha + 0 \cdot 1 & \alpha \cdot 0 + 0 \cdot 1 \\ 1 \cdot \alpha + 1 \cdot 1 & 1 \cdot 0 + 1 \cdot 1 \end{pmatrix} \]
Calculating each element:
- First row, first column: \( \alpha^2 + 0 = \alpha^2 \)
- First row, second column: \( 0 + 0 = 0 \)
- Second row, first column: \( \alpha + 1 \)
- Second row, second column: \( 0 + 1 = 1 \)

Thus, we have:
\[ A^2 = \begin{pmatrix} \alpha^2 & 0 \\ \alpha + 1 & 1 \end{pmatrix} \]

### Step 3: Set \( A^2 \) equal to \( B \)
Now we set \( A^2 \) equal to \( B \):
\[ \begin{pmatrix} \alpha^2 & 0 \\ \alpha + 1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 5 & 1 \end{pmatrix} \]

### Step 4: Create equations from the matrix equality
From the equality of the matrices, we can derive the following equations:
1. \( \alpha^2 = 1 \)
2. \( \alpha + 1 = 5 \)

### Step 5: Solve the equations
**From the first equation:**
\[ \alpha^2 = 1 \implies \alpha = 1 \text{ or } \alpha = -1 \]

**From the second equation:**
\[ \alpha + 1 = 5 \implies \alpha = 4 \]

### Step 6: Check for consistency
We have two values for \( \alpha \): \( 1 \), \( -1 \), and \( 4 \). However, both \( \alpha = 1 \) and \( \alpha = -1 \) do not satisfy the second equation \( \alpha + 1 = 5 \). Therefore, the only value that satisfies the second equation is \( \alpha = 4 \).

### Conclusion
The value of \( \alpha \) for which \( A^2 = B \) is:
\[ \alpha = 4 \]