If`A=[(2,0),(-3,1)] and B=[(4,-3),(-6,2)]` are such that 4A+3X=5B then x=?

Options

A. `[(4,-5),(-6,2)]`
B. `[(4,5),(-6,-2)]`
C. `[(-4,5),(6,-2)]`
D. none of these

Explanation

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

### Step 1: Calculate \( 4A \)

First, we need to multiply matrix \( A \) by 4.

\[ 4A = 4 \times \begin{pmatrix} 2 & 0 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} 4 \times 2 & 4 \times 0 \\ 4 \times -3 & 4 \times 1 \end{pmatrix} = \begin{pmatrix} 8 & 0 \\ -12 & 4 \end{pmatrix} \]

**Hint:** To multiply a matrix by a scalar, multiply each element of the matrix by that scalar.

### Step 2: Calculate \( 5B \)

Next, we will multiply matrix \( B \) by 5.

\[ 5B = 5 \times \begin{pmatrix} 4 & -3 \\ -6 & 2 \end{pmatrix} = \begin{pmatrix} 5 \times 4 & 5 \times -3 \\ 5 \times -6 & 5 \times 2 \end{pmatrix} = \begin{pmatrix} 20 & -15 \\ -30 & 10 \end{pmatrix} \]

**Hint:** Similar to the previous step, multiply each element of matrix \( B \) by 5.

### Step 3: Set up the equation for \( 3X \)

Now, we can rewrite the equation \( 4A + 3X = 5B \) as:

\[ 3X = 5B - 4A \]

Substituting the values we calculated:

\[ 3X = \begin{pmatrix} 20 & -15 \\ -30 & 10 \end{pmatrix} - \begin{pmatrix} 8 & 0 \\ -12 & 4 \end{pmatrix} \]

### Step 4: Perform the subtraction

Now, we will subtract \( 4A \) from \( 5B \):

\[ 3X = \begin{pmatrix} 20 - 8 & -15 - 0 \\ -30 - (-12) & 10 - 4 \end{pmatrix} = \begin{pmatrix} 12 & -15 \\ -30 + 12 & 6 \end{pmatrix} = \begin{pmatrix} 12 & -15 \\ -18 & 6 \end{pmatrix} \]

**Hint:** When subtracting matrices, subtract corresponding elements.

### Step 5: Solve for \( X \)

Now, we need to divide each element of \( 3X \) by 3 to find \( X \):

\[ X = \frac{1}{3} \begin{pmatrix} 12 & -15 \\ -18 & 6 \end{pmatrix} = \begin{pmatrix} \frac{12}{3} & \frac{-15}{3} \\ \frac{-18}{3} & \frac{6}{3} \end{pmatrix} = \begin{pmatrix} 4 & -5 \\ -6 & 2 \end{pmatrix} \]

**Hint:** To find \( X \), divide each element of \( 3X \) by 3.

### Final Answer

Thus, the matrix \( X \) is:

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