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The given problem is:

Matrices \( A \) and \( B \) are:

\[
A = \begin{bmatrix} x & 1 \\ y & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & x \\ -y & -2 \end{bmatrix}
\]

We are tasked to find:
1. The order of the null matrix (i.e., \( AB \) must be the zero matrix).
2. The possible values of \( x \) and \( y \).

### Step 1: Matrix Multiplication of \( A \) and \( B \)

To calculate \( AB \), we perform the multiplication of these two matrices:

\[
AB = \begin{bmatrix} x & 1 \\ y & 2 \end{bmatrix} \begin{bmatrix} 1 & x \\ -y & -2 \end{bmatrix}
\]

Multiplying the matrices:

\[
AB = \begin{bmatrix} (x \cdot 1 + 1 \cdot (-y)) & (x \cdot x + 1 \cdot (-2)) \\ (y \cdot 1 + 2 \cdot (-y)) & (y \cdot x + 2 \cdot (-2)) \end{bmatrix}
\]

Simplifying the elements of \( AB \):

\[
AB = \begin{bmatrix} (x - y) & (x^2 - 2) \\ (y - 2y) & (yx - 4) \end{bmatrix}
\]

\[
AB = \begin{bmatrix} x - y & x^2 - 2 \\ -y & yx - 4 \end{bmatrix}
\]

### Step 2: Null Matrix Condition

For \( AB \) to be a null matrix, all elements of \( AB \) must be zero:

\[
\begin{bmatrix} x - y & x^2 - 2 \\ -y & yx - 4 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
\]

This gives us the following system of equations:
1. \( x - y = 0 \)
2. \( x^2 - 2 = 0 \)
3. \( -y = 0 \)
4. \( yx - 4 = 0 \)

### Step 3: Solve the System of Equations

From equation (3), we have:
\[
y = 0
\]

Substitute \( y = 0 \) into equation (1):
\[
x - 0 = 0 \implies x = 0
\]

Now, check with the remaining equations:
- Equation (2): \( x^2 - 2 = 0 \) gives \( x^2 = 2 \), so \( x = \pm \sqrt{2} \).
- Equation (4): If \( y = 0 \), this becomes \( 0 \cdot x - 4 = 0 \), which is inconsistent.

Thus, there seems to be a contradiction unless \( y \neq 0 \).

### Conclusion

We found that \( x = 0 \) and \( y = 0 \) leads to contradictions. Therefore, valid answers require \( x \approx \pm sqrt

Given matrices A = [[x, 1], [y, 2]] and B = [[1, x], [-y, -2]] such that AB is a null matrix, find the order of the null matrix and possible values of x and y.

Solve the matrix multiplication problem where A and B are matrices, and AB is a null matrix. Learn how to find the order of the null matrix and the possible values of x and y through step-by-step matrix multiplication and solving a system of linear equations.

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