Let ` A= [[5,5alpha,alpha],[0,alpha,5alpha],[0,0,5]]` .` If |A^2| = 25,` then `alpha` equals to:

Options

A. `5^(2)`
B. 1
C. `1//5`
D. 5

Explanation

To solve the problem, we need to find the value of `alpha` such that the determinant of the matrix `A` squared equals 25. Let's break it down step by step.

Given:
\[ A = \begin{bmatrix} 5 & 5\alpha & \alpha \\ 0 & \alpha & 5\alpha \\ 0 & 0 & 5 \end{bmatrix} \]

### Step 1: Calculate the Determinant of Matrix A

To find the determinant of the matrix \( A \), we can use the formula for the determinant of a 3x3 matrix:

\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \( A \):
- \( a = 5 \), \( b = 5\alpha \), \( c = \alpha \)
- \( d = 0 \), \( e = \alpha \), \( f = 5\alpha \)
- \( g = 0 \), \( h = 0 \), \( i = 5 \)

Using the determinant formula, we have:

\[ |A| = 5(\alpha \cdot 5 - 5\alpha \cdot 0) - 5\alpha(0 \cdot 5 - 5\alpha \cdot 0) + \alpha(0 \cdot 0 - \alpha \cdot 0) \]

This simplifies to:

\[ |A| = 5(5\alpha) = 25\alpha \]

### Step 2: Relate the Determinant to the Given Condition

We know from the problem statement that:

\[ |A^2| = 25 \]

Using the property of determinants, we have:

\[ |A^2| = |A|^2 \]

Thus:

\[ |A|^2 = 25 \]

Substituting the determinant we found:

\[ (25\alpha)^2 = 25 \]

### Step 3: Solve for Alpha

Expanding the equation gives:

\[ 625\alpha^2 = 25 \]

Dividing both sides by 625:

\[ \alpha^2 = \frac{25}{625} = \frac{1}{25} \]

Taking the square root of both sides:

\[ \alpha = \pm \frac{1}{5} \]

### Final Answer

Thus, the values of \( \alpha \) are:

\[ \alpha = \frac{1}{5} \quad \text{or} \quad \alpha = -\frac{1}{5} \]