If A = `({:(5, 5x, x ),(0, x , 5x),(0, 0, 5):}) and |A^(2)| = 25 ` , then |x| is equal to

Options

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

Answer

To solve the problem, we need to find the value of |x| given the matrix A and the condition |A²| = 25.

Given:
\[ A = \begin{pmatrix} 5 & 5x & x \\ 0 & x & 5x \\ 0 & 0 & 5 \end{pmatrix} \]

### Step 1: Calculate A²
To find A², we will multiply matrix A by itself.

\[ A^2 = A \cdot A = \begin{pmatrix} 5 & 5x & x \\ 0 & x & 5x \\ 0 & 0 & 5 \end{pmatrix} \cdot \begin{pmatrix} 5 & 5x & x \\ 0 & x & 5x \\ 0 & 0 & 5 \end{pmatrix} \]

Calculating each element of A²:

- First row, first column:
\[ 5 \cdot 5 + 5x \cdot 0 + x \cdot 0 = 25 \]

- First row, second column:
\[ 5 \cdot 5x + 5x \cdot x + x \cdot 0 = 25x + 5x^2 \]

- First row, third column:
\[ 5 \cdot x + 5x \cdot 5x + x \cdot 5 = 5x + 25x^2 + 5x = 10x + 25x^2 \]

- Second row, first column:
\[ 0 \cdot 5 + x \cdot 0 + 5x \cdot 0 = 0 \]

- Second row, second column:
\[ 0 \cdot 5x + x \cdot x + 5x \cdot 0 = x^2 \]

- Second row, third column:
\[ 0 \cdot x + x \cdot 5x + 5x \cdot 5 = 5x^2 + 25x = 30x^2 \]

- Third row, first column:
\[ 0 \cdot 5 + 0 \cdot 0 + 5 \cdot 0 = 0 \]

- Third row, second column:
\[ 0 \cdot 5x + 0 \cdot x + 5 \cdot 0 = 0 \]

- Third row, third column:
\[ 0 \cdot x + 0 \cdot 5x + 5 \cdot 5 = 25 \]

Putting it all together, we get:
\[ A^2 = \begin{pmatrix} 25 & 25x + 5x^2 & 10x + 25x^2 \\ 0 & x^2 & 30x^2 \\ 0 & 0 & 25 \end{pmatrix} \]

### Step 2: Calculate the Determinant of A²
The determinant of a triangular matrix is the product of its diagonal elements. Therefore:
\[ |A^2| = 25 \cdot (x^2) \cdot 25 = 625x^2 \]

### Step 3: Set the Determinant Equal to 25
According to the problem, we have:
\[ |A^2| = 25 \]
Thus:
\[ 625x^2 = 25 \]

### Step 4: Solve for x²
Dividing both sides by 625:
\[ x^2 = \frac{25}{625} = \frac{1}{25} \]

### Step 5: Find |x|
Taking the square root of both sides:
\[ |x| = \sqrt{\frac{1}{25}} = \frac{1}{5} \]

### Final Answer
Thus, the value of |x| is:
\[ |x| = \frac{1}{5} \]