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Write Minors and Cofactors of the elements of following determinants: $(i) \quad \begin{vmatrix} 2&-4 \\ 0&3 \end{vmatrix}$

Note: This is part 1 of a 2 part question, split as 2 separate questions here.
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Toolbox:
• (i)Minor of an element $a_{ij}$ of a determinant is the determinant obtained by deleting it $i^{th}$ row and $j^{th}$ column in which $a_{ij}$ lies.Minor of an element $a_{ij}$ is denoted by $M_{ij}$
• (ii)Cofactor of an element $a_{ij}$ denoted by $A_{ij}$ is defined by $A_{ij}=(-1)^{i+j}M_{ij}$,where $M_{ij}$ is minor of $a_{ij}$
• Write minors and cofactors of the elements of the following determinant:
(i)Given $\begin{vmatrix} 2 & -4\\0 & 3\end{vmatrix}$

We know that minor of the element $a_{ij}$ is $M_{ij}$

Where i is the row and j is the column.

Here $a_{11}=2,$so $M_{11}$=Minor of $a_{11}$ =3

$M_{12}$=Minor of the element $a_{12}=3.$

$M_{21}$=Minor of the element $a_{21}=-4.$

$M_{22}$=Minor of the element $a_{22}=2.$

Now the cofactor of $a_{ij}$ is $A_{ij}$,so

$A_{11}=(-1)^{1+1}\times M_{11}=(-1)^23=3.$

$A_{12}=(-1)^{1+2}\times M_{12}=(-1)^30=0.$

$A_{21}=(-1)^{2+1}\times M_{21}=(-1)^3-4=4.$

$A_{22}=(-1)^{2+2}\times M_{22}=(-1)^42=2.$

answered Feb 23, 2013
edited Feb 24, 2013

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