[0 1 1 ; 1  0 2; 1 2 0] square this matrix please

To square a matrix, we multiply the matrix by itself. Let's denote the given matrix as \( A \):

\[
A = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{bmatrix}
\]

Now, we'll calculate the square of \( A \), which is \( A^2 = A \times A \).

### Step 1: Matrix Multiplication
The element at position \((i, j)\) in the resulting matrix is found by taking the dot product of the \(i\)-th row of the first matrix and the \(j\)-th column of the second matrix.

Let's compute each element of \( A^2 \):

\[
A^2 = \begin{bmatrix}
0 \cdot 0 + 1 \cdot 1 + 1 \cdot 1 & 0 \cdot 1 + 1 \cdot 0 + 1 \cdot 2 & 0 \cdot 1 + 1 \cdot 2 + 1 \cdot 0 \\
1 \cdot 0 + 0 \cdot 1 + 2 \cdot 1 & 1 \cdot 1 + 0 \cdot 0 + 2 \cdot 2 & 1 \cdot 1 + 0 \cdot 2 + 2 \cdot 0 \\
1 \cdot 0 + 2 \cdot 1 + 0 \cdot 1 & 1 \cdot 1 + 2 \cdot 0 + 0 \cdot 2 & 1 \cdot 1 + 2 \cdot 2 + 0 \cdot 0
\end{bmatrix}
\]

### Step 2: Calculate Each Element

- First row:
  - \( (1,1) \): \( 0 \times 0 + 1 \times 1 + 1 \times 1 = 0 + 1 + 1 = 2 \)
  - \( (1,2) \): \( 0 \times 1 + 1 \times 0 + 1 \times 2 = 0 + 0 + 2 = 2 \)
  - \( (1,3) \): \( 0 \times 1 + 1 \times 2 + 1 \times 0 = 0 + 2 + 0 = 2 \)

- Second row:
  - \( (2,1) \): \( 1 \times 0 + 0 \times 1 + 2 \times 1 = 0 + 0 + 2 = 2 \)
  - \( (2,2) \): \( 1 \times 1 + 0 \times 0 + 2 \times 2 = 1 + 0 + 4 = 5 \)
  - \( (2,3) \): \( 1 \times 1 + 0 \times 2 + 2 \times 0 = 1 + 0 + 0 = 1 \)

- Third row:
  - \( (3,1) \): \( 1 \times 0 + 2 \times 1 + 0 \times 1 = 0 + 2 + 0 = 2 \)
  - \( (3,2) \): \( 1 \times 1 + 2 \times 0 + 0 \times 2 = 1 + 0 + 0 = 1 \)
  - \( (3,3) \): \( 1 \times 1 + 2 \times 2 + 0 \times 0 = 1 + 4 + 0 = 5 \)

### Step 3: Write the Result

\[
A^2 = \begin{bmatrix} 2 & 2 & 2 \\ 2 & 5 & 1 \\ 2 & 1 & 5 \end{bmatrix}
\]

This is the squared matrix \( A^2 \).

Would you like further details or have any questions?

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Here are some related questions you might find interesting:
1. What happens when you square a diagonal matrix?
2. How do you compute the determinant of a squared matrix?
3. How does matrix multiplication differ from element-wise multiplication?
4. What are the eigenvalues of the original matrix?
5. How would you interpret squaring a matrix in the context of linear transformations?

**Tip:** Squaring a matrix is not the same as squaring each individual element—it's a specific operation in linear algebra involving matrix multiplication.

Explore how to square a matrix using matrix multiplication with a step-by-step example. This problem involves multiplying a given matrix by itself to find the squared matrix. Suitable for advanced students or anyone interested in linear algebra.

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