If X and Y are the matrices of order 2 `xx` 2 each and `2X-3Y=[{:(-7,0),(7,-13):}]and 3X + 2Y=[{:(9,13),(4,13):}]`, then what is Y equal to ?

Options

A. `[{:(1,3),(-2,1):}]`
B. `[{:(1,3),(2,1):}]`
C. `[{:(3,1),(-1,5):}]`
D. `[{:(3,2),(1,-5):}]`

Explanation

To find the matrix \( Y \), we start with the given equations:

1. \( 2X - 3Y = \begin{pmatrix} -7 & 0 \\ 7 & -13 \end{pmatrix} \) (Equation 1)
2. \( 3X + 2Y = \begin{pmatrix} 9 & 13 \\ 4 & 13 \end{pmatrix} \) (Equation 2)

### Step 1: Multiply the equations to align coefficients

To eliminate one of the variables, we can multiply the first equation by 3 and the second equation by 2, so that the coefficients of \( X \) will be the same.

- Multiply Equation 1 by 3:
\[ 3(2X - 3Y) = 3\begin{pmatrix} -7 & 0 \\ 7 & -13 \end{pmatrix} \]
This gives:
\[ 6X - 9Y = \begin{pmatrix} -21 & 0 \\ 21 & -39 \end{pmatrix} \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 2:
\[ 2(3X + 2Y) = 2\begin{pmatrix} 9 & 13 \\ 4 & 13 \end{pmatrix} \]
This gives:
\[ 6X + 4Y = \begin{pmatrix} 18 & 26 \\ 8 & 26 \end{pmatrix} \quad \text{(Equation 4)} \]

### Step 2: Subtract the two equations

Now, we will subtract Equation 4 from Equation 3 to eliminate \( X \):

\[ (6X - 9Y) - (6X + 4Y) = \begin{pmatrix} -21 & 0 \\ 21 & -39 \end{pmatrix} - \begin{pmatrix} 18 & 26 \\ 8 & 26 \end{pmatrix} \]

This simplifies to:
\[ -9Y - 4Y = \begin{pmatrix} -21 - 18 & 0 - 26 \\ 21 - 8 & -39 - 26 \end{pmatrix} \]
\[ -13Y = \begin{pmatrix} -39 & -26 \\ 13 & -65 \end{pmatrix} \]

### Step 3: Solve for \( Y \)

Now, divide both sides by -13 to isolate \( Y \):

\[ Y = \frac{1}{-13} \begin{pmatrix} -39 & -26 \\ 13 & -65 \end{pmatrix} \]
\[ Y = \begin{pmatrix} 3 & 2 \\ -1 & 5 \end{pmatrix} \]

### Final Result

Thus, the matrix \( Y \) is:
\[ Y = \begin{pmatrix} 3 & 2 \\ -1 & 5 \end{pmatrix} \]

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