If `A=[(3,2),(7,5)], B=[(6,7),(8,9)]` , then adj (AB) is equal to

Options

A. `[(94,-39),(-82,34)]`
B. `[(94,-39),(82,-34)]`
C. `[(94,-82),(-39,34)]`
D. `[(-94,-39),(82,34)]`

Answer

To find the adjoint of the product of two matrices \( A \) and \( B \), we will follow these steps:

### Step 1: Calculate the product \( AB \)

Given:
\[ A = \begin{pmatrix} 3 & 2 \\ 7 & 5 \end{pmatrix}, \quad B = \begin{pmatrix} 6 & 7 \\ 8 & 9 \end{pmatrix} \]

To find \( AB \), we use the formula for matrix multiplication:
\[ AB = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{pmatrix} \]

Calculating each element:
- First row, first column:
\[ 3 \cdot 6 + 2 \cdot 8 = 18 + 16 = 34 \]
- First row, second column:
\[ 3 \cdot 7 + 2 \cdot 9 = 21 + 18 = 39 \]
- Second row, first column:
\[ 7 \cdot 6 + 5 \cdot 8 = 42 + 40 = 82 \]
- Second row, second column:
\[ 7 \cdot 7 + 5 \cdot 9 = 49 + 45 = 94 \]

Thus, we have:
\[ AB = \begin{pmatrix} 34 & 39 \\ 82 & 94 \end{pmatrix} \]

### Step 2: Find the adjoint of \( AB \)

For a \( 2 \times 2 \) matrix:
\[ \text{If } M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \text{ then } \text{adj}(M) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

Applying this to our matrix \( AB \):
\[ AB = \begin{pmatrix} 34 & 39 \\ 82 & 94 \end{pmatrix} \]

We swap \( a \) and \( d \) (34 and 94), and change the signs of \( b \) and \( c \) (39 and 82):
\[ \text{adj}(AB) = \begin{pmatrix} 94 & -39 \\ -82 & 34 \end{pmatrix} \]

### Final Answer:
\[ \text{adj}(AB) = \begin{pmatrix} 94 & -39 \\ -82 & 34 \end{pmatrix} \]

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