NCERT Solutions Class 12 Maths Chapter 4 Ex 4.3 | LearnCBSEHub

Minors & Cofactors

NCERT EXERCISE 4.3 • Q1 — Q5 COMPLETE

💡 The Core Definitions

Minor ($M_{ij}$): Determinant obtained by deleting the $i^{th}$ row and $j^{th}$ column.

Cofactor ($A_{ij}$): Minor with a sign attached: $A_{ij} = (-1)^{i+j} M_{ij}$.

Question 01

Write Minors and Cofactors of (i) $\begin{vmatrix} 2 & -4 \\ 0 & 3 \end{vmatrix}$

Minors:
$M_{11} = 3, \quad M_{12} = 0, \quad M_{21} = -4, \quad M_{22} = 2$

Cofactors ($A_{ij} = (-1)^{i+j}M_{ij}$):
$A_{11} = (-1)^{1+1}(3) = 3$
$A_{12} = (-1)^{1+2}(0) = 0$
$A_{21} = (-1)^{2+1}(-4) = 4$
$A_{22} = (-1)^{2+2}(2) = 2$

Question 02 (ii)

Minors and Cofactors for $\begin{vmatrix} 1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2 \end{vmatrix}$

$M_{11} = \begin{vmatrix} 5 & -1 \\ 1 & 2 \end{vmatrix} = 10 – (-1) = 11 \implies A_{11} = 11$
$M_{12} = \begin{vmatrix} 3 & -1 \\ 0 & 2 \end{vmatrix} = 6 – 0 = 6 \implies A_{12} = -6$
$M_{13} = \begin{vmatrix} 3 & 5 \\ 0 & 1 \end{vmatrix} = 3 – 0 = 3 \implies A_{13} = 3$
… and so on for all 9 elements.

Note: Follow the +/- sign pattern strictly.

Question 03

Using Cofactors of second row, evaluate $\Delta = \begin{vmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix}$

$\Delta = a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23}$

$A_{21} = (-1)^{2+1} \begin{vmatrix} 3 & 8 \\ 2 & 3 \end{vmatrix} = -1(9 – 16) = 7$
$A_{22} = (-1)^{2+2} \begin{vmatrix} 5 & 8 \\ 1 & 3 \end{vmatrix} = 1(15 – 8) = 7$
$A_{23} = (-1)^{2+3} \begin{vmatrix} 5 & 3 \\ 1 & 2 \end{vmatrix} = -1(10 – 3) = -7$

$\Delta = 2(7) + 0(7) + 1(-7) = 14 + 0 – 7 = 7$

Result: 7

Question 04 • Advanced Expansion

Using Cofactors of third column, evaluate $\Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix}$

$A_{13} = (z – y), \quad A_{23} = -(z – x) = (x – z), \quad A_{33} = (y – x)$

$\Delta = yz(z-y) + zx(x-z) + xy(y-x)$
Expanding and factoring: $(x-y)(y-z)(z-x)$

Result: $(x-y)(y-z)(z-x)$

Question 05 • MCQ

Find the correct value for $\Delta$.

The value of a determinant is the sum of the products of elements of any row (or column) with their corresponding cofactors.

Checking the options, only (D) maintains the same indices for elements and cofactors ($a_{i1}$ with $A_{i1}$).

Correct Option: (D) $a_{11}A_{11} + a_{21}A_{21} + a_{31}A_{31}$