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Q1(ii):
Write Minors and Cofactors of the elements of following determinant: (ii) $\begin{vmatrix} a & c \\ b & d \end{vmatrix}$
Solution :
1. Definitions and Theoretical Foundation
For a determinant of order $n$, the minor $M_{ij}$ of an element $a_{ij}$ is the determinant of the $(n-1) \times (n-1)$ matrix obtained by deleting the $i$-th row and $j$-th column. The cofactor $C_{ij}$ is defined as $C_{ij} = (-1)^{i+j} M_{ij}$. [Per Definition of Determinant Minors and Cofactors].
2. Given Matrix Specification
Let the given determinant be $\Delta = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = \begin{vmatrix} a & c \\ b & d \end{vmatrix}$.
- $a_{11} = a$
- $a_{12} = c$
- $a_{21} = b$
- $a_{22} = d$
3. Calculation of Minors ($M_{ij}$)
Minors are calculated by excluding the corresponding row and column of the targeted element:
- $M_{11}$: Delete row 1, column 1. The remaining element is $d$. Thus, $M_{11} = d$.
- $M_{12}$: Delete row 1, column 2. The remaining element is $b$. Thus, $M_{12} = b$.
- $M_{21}$: Delete row 2, column 1. The remaining element is $c$. Thus, $M_{21} = c$.
- $M_{22}$: Delete row 2, column 2. The remaining element is $a$. Thus, $M_{22} = a$.
4. Calculation of Cofactors ($C_{ij}$)
Using the relation $C_{ij} = (-1)^{i+j} M_{ij}$:
- $C_{11} = (-1)^{1+1} M_{11} = (+1) \cdot d = d$
- $C_{12} = (-1)^{1+2} M_{12} = (-1) \cdot b = -b$
- $C_{21} = (-1)^{2+1} M_{21} = (-1) \cdot c = -c$
- $C_{22} = (-1)^{2+2} M_{22} = (+1) \cdot a = a$
5. Tabular Summary of Results
| Element ($a_{ij}$) | Minor ($M_{ij}$) | Cofactor ($C_{ij}$) |
|---|---|---|
| $a$ | $d$ | $d$ |
| $c$ | $b$ | $-b$ |
| $b$ | $c$ | $-c$ |
| $d$ | $a$ | $a$ |
Final Solution: The minors are $M_{11}=d, M_{12}=b, M_{21}=c, M_{22}=a$ and the cofactors are $C_{11}=d, C_{12}=-b, C_{21}=-c, C_{22}=a$.
More Questions from Class 12 Mathematics Determinants EXERCISE 4.3
- Q1(i): Write Minors and Cofactors of the elements of following determinant: (i) $\begin{vmatrix} 2 & -4 \\ 0 & 3 \end{vmatrix}$
- 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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