Which of the following represents the product of the matrices below ?
`[ (4,-2),(3,-6)]xx[(0),(2)]`

Options

A. `[(-4),(-12)]`
B. `[(-12)/(0)]`
C. `[-6]`
D. `[6-12]`

Answer

To find the product of the given matrices, we will follow the standard procedure for matrix multiplication.

We have two matrices:
1. Matrix A = \(\begin{pmatrix} 4 & -2 \\ 3 & -6 \end{pmatrix}\)
2. Matrix B = \(\begin{pmatrix} 0 \\ 2 \end{pmatrix}\)

Matrix A is a \(2 \times 2\) matrix and Matrix B is a \(2 \times 1\) matrix. The product of a \(2 \times 2\) matrix and a \(2 \times 1\) matrix will result in a \(2 \times 1\) matrix.

### Step 1: Multiply the first row of Matrix A with Matrix B

To find the first element of the resulting matrix, we multiply the elements of the first row of Matrix A by the corresponding elements of Matrix B and then sum them up:

\[ \text{First element} = (4 \times 0) + (-2 \times 2) \]
Calculating this gives:
\[ = 0 - 4 = -4 \]

### Step 2: Multiply the second row of Matrix A with Matrix B

Next, we find the second element of the resulting matrix by multiplying the elements of the second row of Matrix A by the corresponding elements of Matrix B and summing them up:

\[ \text{Second element} = (3 \times 0) + (-6 \times 2) \]
Calculating this gives:
\[ = 0 - 12 = -12 \]

### Step 3: Combine the results

Now we can combine the results from Step 1 and Step 2 to form the resulting matrix:

\[ \text{Resulting Matrix} = \begin{pmatrix} -4 \\ -12 \end{pmatrix} \]

Thus, the product of the two matrices is:

\[ \begin{pmatrix} -4 \\ -12 \end{pmatrix} \]

### Final Answer

The product of the matrices is \(\begin{pmatrix} -4 \\ -12 \end{pmatrix}\).

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