If `Delta=|[a_(11),a_(12),a_(13)],[a_(21),a_(22),a_(23)],[a_(31),a_(32),a_(33)]|` and `A_(i j)` is cofactors of `a_(i j)`, then value of `Delta` 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_(31)`
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)`

Explanation

To find the value of the determinant \(\Delta\) given by

\[ \Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} \]

we will use the method of co-factors.

### Step 1: Expand the Determinant
We can expand the determinant along the first row:

\[ \Delta = a_{11} \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12} \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13} \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix} \]

### Step 2: Compute the 2x2 Determinants
Now, we will compute each of the 2x2 determinants:

1. For \(\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix}\):

\[ = a_{22}a_{33} - a_{23}a_{32} \]

2. For \(\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix}\):

\[ = a_{21}a_{33} - a_{23}a_{31} \]

3. For \(\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}\):

\[ = a_{21}a_{32} - a_{22}a_{31} \]

### Step 3: Substitute Back into the Determinant
Now, substituting these back into the expression for \(\Delta\):

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

### Step 4: Rearranging Terms
Rearranging the terms gives us:

\[ \Delta = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} \]

### Step 5: Recognizing Cofactors
Notice that each term corresponds to the cofactor of the respective elements \(a_{11}, a_{12}, a_{13}\). Thus, we can express \(\Delta\) in terms of the cofactors:

\[ \Delta = C_{11} + C_{12} + C_{13} \]

where \(C_{ij}\) are the cofactors of the elements \(a_{ij}\).

### Final Answer
Thus, the value of \(\Delta\) is given by:

\[ \Delta = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \]