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Write Minors and Cofactors of the elements of following determinants: Q : 2 (ii) determinant 1 0 4 3 5 -1 0 1 2

Q : 2      Write Minors and Cofactors of the elements of following determinants:

              \small (ii) \begin{vmatrix} 1 &0 &4 \\ 3 & 5 &-1 \\ 0 &1 &2 \end{vmatrix}

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Given determinant : \begin{vmatrix} 1 &0 &4 \\ 3 & 5 &-1 \\ 0 &1 &2 \end{vmatrix}

Finding Minors: by the definition,

M_{11} = minor of  a_{11} = \begin{vmatrix} 5 &-1 \\ 1 &2 \end{vmatrix} = 11     M_{12} = minor of  a_{12} = \begin{vmatrix} 3 &-1 \\ 0 &2 \end{vmatrix} = 6

M_{13} = minor of  a_{13} = \begin{vmatrix} 3 &5 \\ 0 &1 \end{vmatrix} = 3           M_{21} = minor of  a_{21} = \begin{vmatrix} 0 &4 \\ 1 &2 \end{vmatrix} = -4

M_{22} = minor of  a_{22} = \begin{vmatrix} 1 &4 \\ 0 &2 \end{vmatrix} = 2           M_{23} = minor of  a_{23} = \begin{vmatrix} 1 &0 \\ 0 &1 \end{vmatrix} = 1

M_{31} = minor of  a_{31} = \begin{vmatrix} 0 &4 \\ 5 &-1 \end{vmatrix} = -20 

M_{32} = minor of  a_{32} = \begin{vmatrix} 1 &4 \\ 3 &-1 \end{vmatrix} = -1-12=-13

M_{33} = minor of  a_{33} = \begin{vmatrix} 1 &0 \\ 3 &5 \end{vmatrix} = 5

 

Finding the cofactors:

A_{11}= cofactor of a_{11} = (-1)^{1+1}M_{11} = 11

A_{12}= cofactor of a_{12} = (-1)^{1+2}M_{12} = -6

A_{13}= cofactor of a_{13} = (-1)^{1+3}M_{13} = 3

A_{21}= cofactor of a_{21} = (-1)^{2+1}M_{21} = 4

A_{22}= cofactor of a_{22} = (-1)^{2+2}M_{22} = 2

A_{23}= cofactor of a_{23} = (-1)^{2+3}M_{23} = -1

A_{31}= cofactor of a_{31} = (-1)^{3+1}M_{31} = -20

A_{32}= cofactor of a_{32} = (-1)^{3+2}M_{32} = 13

A_{33}= cofactor of a_{33} = (-1)^{3+3}M_{33} = 5.

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