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Q1(i):
Write Minors and Cofactors of the elements of following determinant: (i) $\begin{vmatrix} 2 & -4 \\ 0 & 3 \end{vmatrix}$
Solution :
Given Determinant and Structural Identification
The given determinant is $A = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = \begin{vmatrix} 2 & -4 \\ 0 & 3 \end{vmatrix}$.
The elements are identified as follows:
- $a_{11} = 2$
- $a_{12} = -4$
- $a_{21} = 0$
- $a_{22} = 3$
Step 1: Calculating Minors ($M_{ij}$)
The minor $M_{ij}$ of an element $a_{ij}$ is the determinant of the $1 \times 1$ matrix obtained by deleting the $i^{th}$ row and $j^{th}$ column [Per definition of minors in determinants].
- $M_{11} = |a_{22}| = |3| = 3$
- $M_{12} = |a_{21}| = |0| = 0$
- $M_{21} = |a_{12}| = |-4| = -4$
- $M_{22} = |a_{11}| = |2| = 2$
Step 2: Calculating Cofactors ($C_{ij}$)
The cofactor $C_{ij}$ of an element $a_{ij}$ is defined as $C_{ij} = (-1)^{i+j} M_{ij}$ [By definition of cofactor expansion].
- $C_{11} = (-1)^{1+1} M_{11} = (-1)^2 (3) = 1 \times 3 = 3$
- $C_{12} = (-1)^{1+2} M_{12} = (-1)^3 (0) = -1 \times 0 = 0$
- $C_{21} = (-1)^{2+1} M_{21} = (-1)^3 (-4) = -1 \times (-4) = 4$
- $C_{22} = (-1)^{2+2} M_{22} = (-1)^4 (2) = 1 \times 2 = 2$
Visual Representation of Matrix Indices
Data Synthesis
| Element | Minor ($M_{ij}$) | Cofactor ($C_{ij}$) |
|---|---|---|
| $a_{11} = 2$ | $3$ | $3$ |
| $a_{12} = -4$ | $0$ | $0$ |
| $a_{21} = 0$ | $-4$ | $4$ |
| $a_{22} = 3$ | $2$ | $2$ |
Final Solution: The minors are $M_{11}=3, M_{12}=0, M_{21}=-4, M_{22}=2$ and the corresponding cofactors are $C_{11}=3, C_{12}=0, C_{21}=4, C_{22}=2$.
More Questions from Class 12 Mathematics Determinants EXERCISE 4.3
- 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 $\Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix}$.
- 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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