Solve `[(1,2),(0,0)] [(x,2),(3,3)]=[(7,0),(1,0)]`, and find the value of x.

Options

A. 2
B. 0
C. 3
D. `-(3)/(2)`

Answer

To solve the matrix equation \(\begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} x & 2 \\ 3 & 3 \end{pmatrix} = \begin{pmatrix} 7 & 0 \\ 1 & 0 \end{pmatrix}\), we will follow these steps:

### Step 1: Multiply the matrices
We start by multiplying the two matrices on the left side. The multiplication of two matrices is done by taking the dot product of the rows of the first matrix with the columns of the second matrix.

The first element of the resulting matrix is calculated as:
\[ 1 \cdot x + 2 \cdot 3 = x + 6 \]

The second element of the first row is:
\[ 1 \cdot 2 + 2 \cdot 3 = 2 + 6 = 8 \]

The first element of the second row is:
\[ 0 \cdot x + 0 \cdot 3 = 0 \]

The second element of the second row is:
\[ 0 \cdot 2 + 0 \cdot 3 = 0 \]

Thus, the product of the matrices is:
\[ \begin{pmatrix} x + 6 & 8 \\ 0 & 0 \end{pmatrix} \]

### Step 2: Set the resulting matrix equal to the given matrix
Now we set the resulting matrix equal to the matrix on the right side of the equation:
\[ \begin{pmatrix} x + 6 & 8 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 7 & 0 \\ 1 & 0 \end{pmatrix} \]

### Step 3: Compare corresponding elements
From the equality of the matrices, we can compare the corresponding elements:

1. From the first row, first column:
\[ x + 6 = 7 \]

2. From the first row, second column:
\[ 8 = 0 \quad \text{(This is not true, indicating we won't use this equation)} \]

3. From the second row, first column:
\[ 0 = 1 \quad \text{(This is also not true)} \]

4. From the second row, second column:
\[ 0 = 0 \quad \text{(This is true)} \]

### Step 4: Solve for \(x\)
Now we solve the equation from the first row:
\[ x + 6 = 7 \]
Subtracting 6 from both sides gives:
\[ x = 7 - 6 = 1 \]

### Final Answer
The value of \(x\) is \(1\).

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