If `A=[a_(ij)]` is a square matrix of order 3 and `A_(ij)` denote cofactor of the element `a_(ij)` in `|A|` then the value of `|A|` is given by

Options

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

Answer

To find the value of the determinant of a square matrix \( A \) of order 3, given that \( A_{ij} \) denotes the cofactor of the element \( a_{ij} \) in the determinant \( |A| \), we can follow these steps:

### Step 1: Define the Matrix
Let \( A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \).

### Step 2: Understand the Cofactor
The cofactor \( A_{ij} \) of an element \( a_{ij} \) is given by \( A_{ij} = (-1)^{i+j} |M_{ij}| \), where \( |M_{ij}| \) is the determinant of the matrix obtained by deleting the \( i \)-th row and \( j \)-th column from \( A \).

### Step 3: Use the Cofactor Expansion
The determinant \( |A| \) can be calculated using the cofactor expansion along any row or column. For example, using the first row:
\[ |A| = a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13} \]

### Step 4: Express the Determinant in Terms of Cofactors
From the cofactor expansion, we can express the determinant as:
\[ |A| = a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13} \]
This means that the determinant is a sum of products of the elements of the first row and their corresponding cofactors.

### Step 5: Relate the Cofactors to the Determinant
By the properties of determinants, we know that the sum of the products of any row (or column) and their corresponding cofactors equals the determinant of the matrix. Therefore, we can conclude:
\[ |A| = a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13} = |A| \]

### Conclusion
Thus, the value of the determinant \( |A| \) can be expressed in terms of its cofactors as:
\[ |A| = a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13} \]