To verify the given statement for the matrix [tex]\(A\)[/tex], we are interested in calculating [tex]\(A \cdot (\operatorname{Adj} A)\)[/tex] and checking whether it equals [tex]\(|A| I\)[/tex], where [tex]\(I\)[/tex] is the identity matrix. We'll proceed through the following steps:
### 1. Calculate the Determinant of [tex]\(A\)[/tex]
Given [tex]\( A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix} \)[/tex]:
[tex]\[
|A| = \text{det}(A) = -11
\][/tex]
### 2. Calculate the Adjugate (Adjoint) of [tex]\(A\)[/tex]
The adjugate of [tex]\(A\)[/tex], denoted as [tex]\(\operatorname{Adj} A\)[/tex], is calculated. The matrix obtained is:
[tex]\[
\operatorname{Adj} A = \begin{bmatrix} 3 & -4 & -5 \\ -9 & 1 & 4 \\ -5 & 3 & 1 \end{bmatrix}
\][/tex]
### 3. Multiply [tex]\(A\)[/tex] by [tex]\(\operatorname{Adj} A\)[/tex]
We now calculate the product [tex]\(A \cdot (\operatorname{Adj} A)\)[/tex]:
[tex]\[
A \cdot (\operatorname{Adj} A) = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix} \cdot \begin{bmatrix} 3 & -4 & -5 \\ -9 & 1 & 4 \\ -5 & 3 & 1 \end{bmatrix}
\][/tex]
Upon calculation, this results in:
[tex]\[
A \cdot (\operatorname{Adj} A) = \begin{bmatrix} -11 & 0 & 0 \\ 0 & -11 & 0 \\ 0 & 0 & -11 \end{bmatrix}
\][/tex]
### 4. Form the Determinant Times the Identity Matrix
We multiply the determinant [tex]\(|A| = -11\)[/tex] by the identity matrix of size 3:
[tex]\[
|A| I = -11 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} -11 & 0 & 0 \\ 0 & -11 & 0 \\ 0 & 0 & -11 \end{bmatrix}
\][/tex]
### 5. Verify the Equality
Compare the results from steps 3 and 4:
[tex]\[
A \cdot (\operatorname{Adj} A) = -11 I = \begin{bmatrix} -11 & 0 & 0 \\ 0 & -11 & 0 \\ 0 & 0 & -11 \end{bmatrix}
\][/tex]
### Conclusion
From the calculations, we can verify that:
[tex]\[
A \cdot (\operatorname{Adj} A) = |A| I
\][/tex]
Thus, the given statement is confirmed: [tex]\(A \cdot (\operatorname{Adj} A) = |A| I\)[/tex].
