If `A=[(1,0,1),(0,0,0),(1,0,1)]`, then find `|A^(2)|`.

Answer

To find \( |A^2| \) where \( A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix \( A \)

The determinant of a 3x3 matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \) is given by the formula:

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

For our matrix \( A \):

\[ A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{pmatrix} \]

Substituting the values into the determinant formula:

\[ |A| = 1(0 \cdot 1 - 0 \cdot 0) - 0(0 \cdot 1 - 1 \cdot 0) + 1(0 \cdot 0 - 0 \cdot 1) \]

This simplifies to:

\[ |A| = 1(0 - 0) - 0(0 - 0) + 1(0 - 0) = 0 \]

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

Using the property of determinants, we know that:

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

Since we have already found \( |A| = 0 \):

\[ |A^2| = 0^2 = 0 \]

### Final Answer

Thus, the value of \( |A^2| \) is:

\[ \boxed{0} \]