If `A=[{:(a_(11),a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33)):}]` and `A_(ij)` is co-factos of `a_(ij)`, then A is given by :

Options

A. `a_(11)A_(31)+a_(12)A_(32)+a_(13)A_(33)`
B. `a_(11)A_(11)+a_(12)A_(21)+a_(13)A_(33)`
C. `a_(21)A_(11)+a_(22)A_(12)+a_(23)A_(13)`
D. `a_(11)A_(11)+a_(21)A_(21)+a_(31)A_(31)`

Answer

To find the determinant of the matrix \( A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \), we will use the formula for the determinant of a 3x3 matrix.

### Step 1: Write the formula for the determinant of a 3x3 matrix.
The determinant of a 3x3 matrix can be calculated using the following formula:

\[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]

### Step 2: Apply the formula to our matrix.
Using the elements of matrix \( A \):

\[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]

### Step 3: Simplify the expression.
Now, we can simplify the expression to get the determinant in a more compact form:

\[ \text{det}(A) = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33} \]

### Step 4: Conclusion.
Thus, the determinant of the matrix \( A \) is given by:

\[ \text{det}(A) = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33} \]