Let's solve the problem step by step.
First, we are given the matrix [tex]\( A \)[/tex]:
[tex]\[ A = \begin{pmatrix} 5 & 3 \\ 4 & 2 \end{pmatrix} \][/tex]
To find [tex]\( A \cdot (\operatorname{Adj} A) \)[/tex], we need to follow these steps:
### 1. Find the adjugate of [tex]\( A \)[/tex]:
The adjugate (or adjoint) of a matrix is the transpose of the cofactor matrix. For a [tex]\( 2 \times 2 \)[/tex] matrix [tex]\( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex], the adjugate is defined as:
[tex]\[ \operatorname{Adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \][/tex]
Applying this to our matrix [tex]\( A \)[/tex]:
[tex]\[ A = \begin{pmatrix} 5 & 3 \\ 4 & 2 \end{pmatrix} \][/tex]
The cofactor matrix is:
[tex]\[ \text{Cof}(A) = \begin{pmatrix} 2 & -3 \\ -4 & 5 \end{pmatrix} \][/tex]
Taking the transpose of the cofactor matrix gives us the adjugate:
[tex]\[ \operatorname{Adj}(A) = \begin{pmatrix} 2 & -4 \\ -3 & 5 \end{pmatrix} \][/tex]
### 2. Calculate [tex]\( A \cdot (\operatorname{Adj} A) \)[/tex]:
Now we need to multiply [tex]\( A \)[/tex] by its adjugate.
[tex]\[ A = \begin{pmatrix} 5 & 3 \\ 4 & 2 \end{pmatrix} \][/tex]
[tex]\[ \operatorname{Adj}(A) = \begin{pmatrix} 2 & -4 \\ -3 & 5 \end{pmatrix} \][/tex]
Performing the matrix multiplication:
[tex]\[ A \cdot (\operatorname{Adj}(A)) = \begin{pmatrix} 5 & 3 \\ 4 & 2 \end{pmatrix} \begin{pmatrix} 2 & -4 \\ -3 & 5 \end{pmatrix} \][/tex]
Let's compute each element of the resulting matrix:
- First row, first column:
[tex]\[ (5 \cdot 2) + (3 \cdot -3) = 10 - 9 = 1 \][/tex]
- First row, second column:
[tex]\[ (5 \cdot -4) + (3 \cdot 5) = -20 + 15 = -5 \][/tex]
- Second row, first column:
[tex]\[ (4 \cdot 2) + (2 \cdot -3) = 8 - 6 = 2 \][/tex]
- Second row, second column:
[tex]\[ (4 \cdot -4) + (2 \cdot 5) = -16 + 10 = -6 \][/tex]
Thus, the product [tex]\( A \cdot (\operatorname{Adj} A) \)[/tex] is:
[tex]\[ A \cdot (\operatorname{Adj} A) = \begin{pmatrix} 1 & -5 \\ 2 & -6 \end{pmatrix} \][/tex]
### Summary:
The final result of the matrix multiplication [tex]\( A \cdot (\operatorname{Adj} A) \)[/tex] is:
[tex]\[ \begin{pmatrix} 1 & -5 \\ 2 & -6 \end{pmatrix} \][/tex]
So, we have:
[tex]\[ A \cdot (\operatorname{Adj} A) = \begin{pmatrix} 1 & -5 \\ 2 & -6 \end{pmatrix} \][/tex]
