If `({:(1, 0),(0, 1):})({:(a, b+c),(b-c, d):}) = ({:(4, -5),(3, 2):})`, then (a-b) + (c-d)=____.

Options

A. -2
B. 9
C. 2
D. -1

Explanation

To solve the equation given by the matrix multiplication, we need to break it down step by step.

Given:
\[ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} a & b+c \\ b-c & d \end{pmatrix} = \begin{pmatrix} 4 & -5 \\ 3 & 2 \end{pmatrix} \]

### Step 1: Perform the Matrix Multiplication

The multiplication of the two matrices on the left side is done as follows:

\[ \begin{pmatrix} 1 \cdot a + 0 \cdot (b-c) & 1 \cdot (b+c) + 0 \cdot d \\ 0 \cdot a + 1 \cdot (b-c) & 0 \cdot (b+c) + 1 \cdot d \end{pmatrix} = \begin{pmatrix} a & b+c \\ b-c & d \end{pmatrix} \]

So, we have:
\[ \begin{pmatrix} a & b+c \\ b-c & d \end{pmatrix} = \begin{pmatrix} 4 & -5 \\ 3 & 2 \end{pmatrix} \]

### Step 2: Set Up the Equations

From the equality of the matrices, we can set up the following equations:

1. \( a = 4 \)
2. \( b + c = -5 \)
3. \( b - c = 3 \)
4. \( d = 2 \)

### Step 3: Solve for \( b \) and \( c \)

We have two equations involving \( b \) and \( c \):

1. \( b + c = -5 \) (Equation 1)
2. \( b - c = 3 \) (Equation 2)

To solve these equations, we can add Equation 1 and Equation 2:

\[ (b + c) + (b - c) = -5 + 3 \]
\[ 2b = -2 \implies b = -1 \]

Now, substitute \( b = -1 \) back into Equation 1 to find \( c \):

\[ -1 + c = -5 \implies c = -5 + 1 = -4 \]

### Step 4: Find \( a - b + c - d \)

Now we have:
- \( a = 4 \)
- \( b = -1 \)
- \( c = -4 \)
- \( d = 2 \)

Now we calculate \( (a - b) + (c - d) \):

\[ a - b + c - d = 4 - (-1) + (-4) - 2 \]
\[ = 4 + 1 - 4 - 2 \]
\[ = 5 - 6 = -1 \]

### Final Answer

Thus, the value of \( (a - b) + (c - d) \) is:
\[ \boxed{-1} \]