If `P=[{:(,2,6),(,3,9):}] and Q=[{:(,3,x),(,y,2):}]`. find x and y such that PQ=null matrix.

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) such that the product of matrices \( P \) and \( Q \) results in a null matrix (a matrix where all elements are zero).

Given:
\[ P = \begin{pmatrix} 2 & 6 \\ 3 & 9 \end{pmatrix} \]
\[ Q = \begin{pmatrix} 3 & x \\ y & 2 \end{pmatrix} \]

### Step 1: Multiply the matrices \( P \) and \( Q \)

The product \( PQ \) is calculated as follows:
\[ PQ = \begin{pmatrix} 2 & 6 \\ 3 & 9 \end{pmatrix} \begin{pmatrix} 3 & x \\ y & 2 \end{pmatrix} \]

Using the rules of matrix multiplication, we calculate each element of the resulting matrix.

### Step 2: Calculate the elements of the resulting matrix

1. **First Row, First Column:**
\[ 2 \cdot 3 + 6 \cdot y = 6 + 6y \]

2. **First Row, Second Column:**
\[ 2 \cdot x + 6 \cdot 2 = 2x + 12 \]

3. **Second Row, First Column:**
\[ 3 \cdot 3 + 9 \cdot y = 9 + 9y \]

4. **Second Row, Second Column:**
\[ 3 \cdot x + 9 \cdot 2 = 3x + 18 \]

So, the product \( PQ \) can be expressed as:
\[ PQ = \begin{pmatrix} 6 + 6y & 2x + 12 \\ 9 + 9y & 3x + 18 \end{pmatrix} \]

### Step 3: Set the resulting matrix equal to the null matrix

Since we want \( PQ \) to be a null matrix, we set each element equal to zero:
\[ \begin{pmatrix} 6 + 6y & 2x + 12 \\ 9 + 9y & 3x + 18 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

This gives us the following equations:
1. \( 6 + 6y = 0 \)
2. \( 2x + 12 = 0 \)
3. \( 9 + 9y = 0 \)
4. \( 3x + 18 = 0 \)

### Step 4: Solve the equations

**From the first equation:**
\[ 6 + 6y = 0 \implies 6y = -6 \implies y = -1 \]

**From the second equation:**
\[ 2x + 12 = 0 \implies 2x = -12 \implies x = -6 \]

### Final Answer

Thus, the values of \( x \) and \( y \) are:
\[ x = -6, \quad y = -1 \]