If `A=[(1,2,1),(3,2,3),(2,1,2)]` , then `a_(11) A_(11)+a_(21) A_(21)+a_(31) A_(31)=`

Options

A. 1
B. 2
C. |A|
D. 3

Answer

To solve the problem, we need to compute the expression \( a_{11} A_{11} + a_{21} A_{21} + a_{31} A_{31} \) for the matrix \( A = \begin{pmatrix} 1 & 2 & 1 \\ 3 & 2 & 3 \\ 2 & 1 & 2 \end{pmatrix} \).

### Step 1: Identify the elements \( a_{11}, a_{21}, a_{31} \)
From the matrix \( A \):
- \( a_{11} = 1 \) (element in the first row, first column)
- \( a_{21} = 3 \) (element in the second row, first column)
- \( a_{31} = 2 \) (element in the third row, first column)

### Step 2: Calculate \( A_{11} \)
The minor \( A_{11} \) is obtained by deleting the first row and first column of matrix \( A \):
\[ A_{11} = \begin{vmatrix} 2 & 3 \\ 1 & 2 \end{vmatrix} = (2 \cdot 2) - (3 \cdot 1) = 4 - 3 = 1 \]

### Step 3: Calculate \( A_{21} \)
The minor \( A_{21} \) is obtained by deleting the second row and first column of matrix \( A \):
\[ A_{21} = \begin{vmatrix} 2 & 1 \\ 1 & 2 \end{vmatrix} = (2 \cdot 2) - (1 \cdot 1) = 4 - 1 = 3 \]

### Step 4: Calculate \( A_{31} \)
The minor \( A_{31} \) is obtained by deleting the third row and first column of matrix \( A \):
\[ A_{31} = \begin{vmatrix} 2 & 1 \\ 2 & 3 \end{vmatrix} = (2 \cdot 3) - (1 \cdot 2) = 6 - 2 = 4 \]

### Step 5: Compute the expression
Now, we substitute the values we found into the expression \( a_{11} A_{11} + a_{21} A_{21} + a_{31} A_{31} \):
\[ = 1 \cdot 1 + 3 \cdot 3 + 2 \cdot 4 \]
\[ = 1 + 9 + 8 \]
\[ = 18 \]

### Final Answer
Thus, the value of \( a_{11} A_{11} + a_{21} A_{21} + a_{31} A_{31} \) is \( 18 \).

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