If `[{:(3,0),(0,2):}][{:(x),(y):}] = [{:(3),(2):}]`, then x=1 and y=-1. True or False.

Answer

To solve the equation given in the problem, we need to multiply the two matrices and set them equal to the resulting matrix. Let's break it down step by step.

### Step 1: Write down the matrices
We have two matrices:
1. Matrix A:
\[ A = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix} \]
2. Matrix B:
\[ B = \begin{pmatrix} x \\ y \end{pmatrix} \]
3. Resulting Matrix C:
\[ C = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \]

### Step 2: Multiply the matrices
Now, we multiply matrix A by matrix B:
\[ A \cdot B = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} \]

The multiplication results in:
- The first element: \(3 \cdot x + 0 \cdot y = 3x\)
- The second element: \(0 \cdot x + 2 \cdot y = 2y\)

Thus, we have:
\[ A \cdot B = \begin{pmatrix} 3x \\ 2y \end{pmatrix} \]

### Step 3: Set the resulting matrix equal to matrix C
Now, we set the result equal to matrix C:
\[ \begin{pmatrix} 3x \\ 2y \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \]

### Step 4: Create equations from the matrix equality
From the equality of the matrices, we can derive two equations:
1. \(3x = 3\)
2. \(2y = 2\)

### Step 5: Solve for x and y
Now, we solve each equation:
1. From \(3x = 3\):
\[ x = \frac{3}{3} = 1 \]

2. From \(2y = 2\):
\[ y = \frac{2}{2} = 1 \]

### Conclusion
We find that \(x = 1\) and \(y = 1\).

### Final Statement
The statement "x=1 and y=-1" is **False** because we found \(y = 1\).

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