Solve the matrix equation `[(2,1),(5,0)]-3X=[(-7,4),(2,6)]`

Options

A. `X=[(3,1),(-1,2)]`
B. `X=[(3,-1),(1,-2)]`
C. `X=[(3,-1),(-1,-2)]`
D. `X=[(-3,-1),(1,-2)]`

Answer

To solve the matrix equation \(\begin{pmatrix} 2 & 1 \\ 5 & 0 \end{pmatrix} - 3X = \begin{pmatrix} -7 & 4 \\ 2 & 6 \end{pmatrix}\), we will follow these steps:

### Step 1: Isolate the term involving \(X\)
We start by adding \(3X\) to both sides of the equation:

\[ \begin{pmatrix} 2 & 1 \\ 5 & 0 \end{pmatrix} = 3X + \begin{pmatrix} -7 & 4 \\ 2 & 6 \end{pmatrix} \]

Next, we subtract \(\begin{pmatrix} -7 & 4 \\ 2 & 6 \end{pmatrix}\) from both sides:

\[ \begin{pmatrix} 2 & 1 \\ 5 & 0 \end{pmatrix} - \begin{pmatrix} -7 & 4 \\ 2 & 6 \end{pmatrix} = 3X \]

### Step 2: Perform the matrix subtraction
Now, we perform the subtraction of the matrices:

\[ \begin{pmatrix} 2 - (-7) & 1 - 4 \\ 5 - 2 & 0 - 6 \end{pmatrix} = 3X \]

Calculating each element:

- First row, first column: \(2 + 7 = 9\)
- First row, second column: \(1 - 4 = -3\)
- Second row, first column: \(5 - 2 = 3\)
- Second row, second column: \(0 - 6 = -6\)

Thus, we have:

\[ \begin{pmatrix} 9 & -3 \\ 3 & -6 \end{pmatrix} = 3X \]

### Step 3: Solve for \(X\)
To find \(X\), we need to divide each element of the resulting matrix by 3:

\[ X = \frac{1}{3} \begin{pmatrix} 9 & -3 \\ 3 & -6 \end{pmatrix} \]

Calculating each element:

- First row, first column: \(\frac{9}{3} = 3\)
- First row, second column: \(\frac{-3}{3} = -1\)
- Second row, first column: \(\frac{3}{3} = 1\)
- Second row, second column: \(\frac{-6}{3} = -2\)

Thus, we have:

\[ X = \begin{pmatrix} 3 & -1 \\ 1 & -2 \end{pmatrix} \]

### Final Answer
The solution for the matrix \(X\) is:

\[ X = \begin{pmatrix} 3 & -1 \\ 1 & -2 \end{pmatrix} \]