Write the minors and cofactors of the elements of the following determinant: $\left|\begin{array}{ll}a & c \\ b & d\end{array}\right|$
The given determinant is $\left|\begin{array}{ll}a & c \\ b & d\end{array}\right|$.
The minor of an element $a_{ij}$ is denoted by $M_{ij}$.
$M_{11} = \text{minor of element } a_{11} = d$
$M_{12} = \text{minor of element } a_{12} = b$
$M_{21} = \text{minor of element } a_{21} = c$
$M_{22} = \text{minor of element } a_{22} = a$
The cofactor of an element $a_{ij}$ is denoted by $A_{ij} = (-1)^{i+j} M_{ij}$.
$A_{11} = (-1)^{1+1} M_{11} = (-1)^{2}(d) = d$
$A_{12} = (-1)^{1+2} M_{12} = (-1)^{3}(b) = -b$
$A_{21} = (-1)^{2+1} M_{21} = (-1)^{3}(c) = -c$
$A_{22} = (-1)^{2+2} M_{22} = (-1)^{4}(a) = a$
The minor of an element $a_{ij}$ is denoted by $M_{ij}$.
$M_{11} = \text{minor of element } a_{11} = d$
$M_{12} = \text{minor of element } a_{12} = b$
$M_{21} = \text{minor of element } a_{21} = c$
$M_{22} = \text{minor of element } a_{22} = a$
The cofactor of an element $a_{ij}$ is denoted by $A_{ij} = (-1)^{i+j} M_{ij}$.
$A_{11} = (-1)^{1+1} M_{11} = (-1)^{2}(d) = d$
$A_{12} = (-1)^{1+2} M_{12} = (-1)^{3}(b) = -b$
$A_{21} = (-1)^{2+1} M_{21} = (-1)^{3}(c) = -c$
$A_{22} = (-1)^{2+2} M_{22} = (-1)^{4}(a) = a$
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Similar Questions
Match the following elements of the matrix $A = \left[\begin{array}{ccc} 1 & -1 & 0 \\ 0 & 4 & 2 \\ 3 & -4 & 6 \end{array}\right]$ with their co-factors and choose the correct answer.
Element Co-factor $A$. $-1$ $(1)$ $-2$ $B$. $1$ $(2)$ $32$ $C$. $3$ $(3)$ $4$ $D$. $6$ $(4)$ $6$ $(5)$ $-6$
| Element | Co-factor |
| $A$. $-1$ | $(1)$ $-2$ |
| $B$. $1$ | $(2)$ $32$ |
| $C$. $3$ | $(3)$ $4$ |
| $D$. $6$ | $(4)$ $6$ |
| $(5)$ $-6$ |
Let ${\Delta _1} = \begin{vmatrix} {a_1} & {b_1} & {c_1} \\ {a_2} & {b_2} & {c_2} \\ {a_3} & {b_3} & {c_3} \end{vmatrix}$ and ${\Delta _2} = \begin{vmatrix} {\alpha _1} & {\beta _1} & {\gamma _1} \\ {\alpha _2} & {\beta _2} & {\gamma _2} \\ {\alpha _3} & {\beta _3} & {\gamma _3} \end{vmatrix}$. Then ${\Delta _1} \times {\Delta _2}$ can be expressed as the sum of how many determinants?
Medium
View SolutionThe co-factors of the elements of the second column of $\begin{bmatrix} 1 & -1 & 2 \\ 3 & 2 & 1 \\ -1 & 3 & 4 \end{bmatrix}$ are:
If $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 1 & -3 \\ -1 & 2 & 3 \end{bmatrix}$,then find the value of $A_{31} + A_{32} + A_{33}$,where $A_{ij}$ denotes the cofactor of the element $a_{ij}$ of matrix $A$.
If in the determinant $\Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$,$A_1, B_1, C_1$ etc. are the co-factors of $a_1, b_1, c_1$ etc.,then which of the following relations is incorrect?
Easy
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