If matrix `X=[{:(,-3,4),(,2,-3):}] [{:(,2),(,-2):}] and 2X-3Y=[{:(,10),(,-8):}]` find the matrix 'X' and matrix Y.

Answer

To solve the problem, we need to find matrices \( X \) and \( Y \) given the equation \( 2X - 3Y = \begin{pmatrix} 10 \\ -8 \end{pmatrix} \) and the matrix \( X \) defined as:

\[ X = \begin{pmatrix} -3 & 4 \\ 2 & -3 \end{pmatrix} \]

and the matrix \( Y \) defined as:

\[ Y = \begin{pmatrix} 2 \\ -2 \end{pmatrix} \]

### Step 1: Calculate \( 2X \)

First, we need to calculate \( 2X \):

\[ 2X = 2 \cdot \begin{pmatrix} -3 & 4 \\ 2 & -3 \end{pmatrix} = \begin{pmatrix} 2 \cdot -3 & 2 \cdot 4 \\ 2 \cdot 2 & 2 \cdot -3 \end{pmatrix} = \begin{pmatrix} -6 & 8 \\ 4 & -6 \end{pmatrix} \]

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

### Step 2: Calculate \( 3Y \)

Next, we calculate \( 3Y \):

\[ 3Y = 3 \cdot \begin{pmatrix} 2 \\ -2 \end{pmatrix} = \begin{pmatrix} 3 \cdot 2 \\ 3 \cdot -2 \end{pmatrix} = \begin{pmatrix} 6 \\ -6 \end{pmatrix} \]

**Hint:** Similar to the previous step, multiply each element of the matrix by the scalar.

### Step 3: Substitute into the equation \( 2X - 3Y = \begin{pmatrix} 10 \\ -8 \end{pmatrix} \)

Now we substitute \( 2X \) and \( 3Y \) into the equation:

\[ 2X - 3Y = \begin{pmatrix} -6 & 8 \\ 4 & -6 \end{pmatrix} - \begin{pmatrix} 6 \\ -6 \end{pmatrix} \]

### Step 4: Perform the subtraction

Now we perform the subtraction:

\[ 2X - 3Y = \begin{pmatrix} -6 - 6 \\ 4 - (-6) \end{pmatrix} = \begin{pmatrix} -12 \\ 4 + 6 \end{pmatrix} = \begin{pmatrix} -12 \\ 10 \end{pmatrix} \]

### Step 5: Set the result equal to the given matrix

We know from the problem statement that:

\[ 2X - 3Y = \begin{pmatrix} 10 \\ -8 \end{pmatrix} \]

Thus, we have:

\[ \begin{pmatrix} -12 \\ 10 \end{pmatrix} = \begin{pmatrix} 10 \\ -8 \end{pmatrix} \]

### Step 6: Solve for \( Y \)

Now we need to isolate \( Y \):

Rearranging the equation gives us:

\[ 3Y = 2X - \begin{pmatrix} 10 \\ -8 \end{pmatrix} \]

Substituting \( 2X \):

\[ 3Y = \begin{pmatrix} -12 \\ 10 \end{pmatrix} - \begin{pmatrix} 10 \\ -8 \end{pmatrix} = \begin{pmatrix} -12 - 10 \\ 10 - (-8) \end{pmatrix} = \begin{pmatrix} -22 \\ 18 \end{pmatrix} \]

### Step 7: Divide by 3 to find \( Y \)

Now we divide by 3:

\[ Y = \frac{1}{3} \begin{pmatrix} -22 \\ 18 \end{pmatrix} = \begin{pmatrix} -\frac{22}{3} \\ 6 \end{pmatrix} \]

### Final Result

Thus, the matrices are:

\[ X = \begin{pmatrix} -3 & 4 \\ 2 & -3 \end{pmatrix}, \quad Y = \begin{pmatrix} -\frac{22}{3} \\ 6 \end{pmatrix} \]