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Q4:
Using Cofactors of elements of third column, evaluate $\Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix}$.
Solution :
Given Determinant and Structural Analysis
We are given the determinant:
$\Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix}$
The objective is to evaluate $\Delta$ by expanding along the third column ($C_3$). The elements of $C_3$ are $a_{13} = yz$, $a_{23} = zx$, and $a_{33} = xy$.
Step 1: Calculating Cofactors of the Third Column
The cofactor $C_{ij}$ of an element $a_{ij}$ is defined as $(-1)^{i+j} M_{ij}$, where $M_{ij}$ is the minor obtained by deleting the $i$-th row and $j$-th column. [Per definition of Cofactor in Determinant Theory].
For $a_{13} = yz$:
$C_{13} = (-1)^{1+3} \begin{vmatrix} 1 & y \\ 1 & z \end{vmatrix} = (1)(z - y) = z - y$
For $a_{23} = zx$:
$C_{23} = (-1)^{2+3} \begin{vmatrix} 1 & x \\ 1 & z \end{vmatrix} = -(1)(z - x) = x - z$
For $a_{33} = xy$:
$C_{33} = (-1)^{3+3} \begin{vmatrix} 1 & x \\ 1 & y \end{vmatrix} = (1)(y - x) = y - x$
Step 2: Expansion along the Third Column
The determinant is evaluated as $\Delta = a_{13}C_{13} + a_{23}C_{23} + a_{33}C_{33}$. [Per Laplace Expansion Theorem].
$\Delta = (yz)(z - y) + (zx)(x - z) + (xy)(y - x)$
Step 3: Algebraic Simplification
Expanding the terms:
$\Delta = yz^2 - y^2z + zx^2 - z^2x + xy^2 - x^2y$
Group terms to facilitate factorization:
$\Delta = -(x^2y - xy^2) + (zx^2 - z^2x) + (yz^2 - y^2z)$
$\Delta = -xy(x - y) + zx(x - z) + yz(z - y)$
Continuing the expansion to reach the standard cyclic form:
$\Delta = yz^2 - y^2z + zx^2 - z^2x + xy^2 - x^2y$
Rearranging for factoring: $(x-y)(y-z)(z-x)$
$\Delta = x^2y - x^2z - xy^2 + y^2z + xz^2 - yz^2$
Factoring step-by-step results in: $\Delta = -(x - y)(y - z)(z - x)$
Visual Representation of Matrix Components
Final Solution: $\Delta = -(x - y)(y - z)(z - x) = (x - y)(y - z)(x - z)$ or $(x - y)(y - z)(z - x)$ expressed as $-(x-y)(y-z)(z-x)$
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 $\Delta = \begin{vmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix}$.
- 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
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