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Q5:
If $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$ and $A_{ij}$ is Cofactors of $a_{ij}$, then value of $\Delta$ is given by
Solution :
Theoretical Foundation: Laplace Expansion of a Determinant
The value of a determinant of order 3, denoted as $\Delta$, is defined as the sum of the products of the elements of any row (or column) and their corresponding cofactors. Let $A_{ij}$ represent the cofactor of the element $a_{ij}$. The cofactor $A_{ij}$ is defined as $(-1)^{i+j} M_{ij}$, where $M_{ij}$ is the minor of the element $a_{ij}$.
Step 1: Formal Definition of the Determinant
By the Laplace Expansion theorem, the value of the determinant $\Delta$ can be computed by expanding along any row $i$ or any column $j$. Specifically, for any row $i$ ($i=1, 2, 3$):
$\Delta = \sum_{j=1}^{3} a_{ij} A_{ij}$
Step 2: Explicit Expansion for Each Row
Based on the structure of the determinant $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$, the value is derived as follows:
| Expansion Reference | Mathematical Expression |
|---|---|
| Expanding along Row 1 | $\Delta = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}$ |
| Expanding along Row 2 | $\Delta = a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23}$ |
| Expanding along Row 3 | $\Delta = a_{31}A_{31} + a_{32}A_{32} + a_{33}A_{33}$ |
Analytical Note: Properties of Cofactors
It is important to distinguish the expansion formula from the property of cofactors where the sum of products of elements of one row with the cofactors of another row is zero [Per the properties of determinants in Linear Algebra]. Specifically, if $i \neq k$, then $a_{i1}A_{k1} + a_{i2}A_{k2} + a_{i3}A_{k3} = 0$.
Visual Representation of Matrix Indices
Final Solution: The value of $\Delta$ is given by the sum of the product of elements of any row (or column) and their corresponding cofactors: $\Delta = a_{i1}A_{i1} + a_{i2}A_{i2} + a_{i3}A_{i3}$ for any row $i = 1, 2, \text{ or } 3$.
More Questions from Class 12 Mathematics Determinants EXERCISE 4.3
- Q1(i): Write Minors and Cofactors of the elements of following determinant: (i) $\begin{vmatrix} 2 & -4 \\ 0 & 3 \end{vmatrix}$
- Q1(ii): Write Minors and Cofactors of the elements of following determinant: (ii) $\begin{vmatrix} a & c \\ b & d \end{vmatrix}$
- Q2(i): Write Minors and Cofactors of the elements of following determinant: (i) $\begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix}$
- Q2(ii): Write Minors and Cofactors of the elements of following determinant: (ii) $\begin{vmatrix} 1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2 \end{vmatrix}$
- Q3: Using Cofactors of elements of second row, evaluate $\Delta = \begin{vmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix}$.
- Q4: Using Cofactors of elements of third column, evaluate $\Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix}$.
CBSE Solutions for Class 12 Mathematics Determinants
Chapters in CBSE - Class 12 Mathematics
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