Question

Write Minors and Cofactors of the elements of following determinants:
I. \(\begin{vmatrix}1&0&0\\0&1&0\\0&0&1\end{vmatrix}\)II. \(\begin{vmatrix}1&0&4\\3&5&-1\\0&1&2\end{vmatrix}\)

Answer 1

I. The given determinant is \(\begin{vmatrix}1&0&0\\0&1&0\\0&0&1\end{vmatrix}\)
By the definition of minors and cofactors, we have:
M₁₁ = minor of a₁₁=\(\begin{vmatrix}1&0\\0&1\end{vmatrix}\)=1

M₁₂ = minor of a₁₂=\(\begin{vmatrix}0&0\\0&1\end{vmatrix}\)=0

M₁₃ = minor of a₁₃ =\(\begin{vmatrix}0&1\\0&0\end{vmatrix}\)=0

M₂₁ = minor of a₂₁ =\(\begin{vmatrix}0&0\\0&1\end{vmatrix}\)=0

M₂₂ = minor of a₂₂ =\(\begin{vmatrix}1&0\\0&1\end{vmatrix}\)=1

M₂₃ = minor of a₂₃ =\(\begin{vmatrix}1&0\\0&0\end{vmatrix}\)=0

M₃₁ = minor of a₃₁=\(\begin{vmatrix}0&0\\1&0\end{vmatrix}\)=0

M₃₂ = minor of a₃₂ =\(\begin{vmatrix}1&0\\0&0\end{vmatrix}\)=0

M₃₃ = minor of a₃₃ =\(\begin{vmatrix}1&0\\0&1\end{vmatrix}\)=1

A₁₁ = cofactor of a₁₁\(= (−1)^{1+1} M_{11} = 1\)
A₁₂ = cofactor of a₁₂ \(= (−1)^{1+2} M_{12} = 0\)
A₁₃ = cofactor of a₁₃ \(= (−1)^{1+3} M_{13} = 0\)
A₂₁ = cofactor of a₂₁ \(= (−1)^{2+1} M_{21} = 0\)
A₂₂ = cofactor of a₂₂ \(= (−1)^{2+2} M_{22} = 1\)
A₂₃ = cofactor of a₂₃ \(= (−1)^{2+3} M_{23} = 0\)
A₃₁ = cofactor of a₃₁ \(= (−1)^{3+1} M_{31} = 0\)
A₃₂ = cofactor of a₃₂ \(= (−1)^{3+2} M_{32} = 0\)
A₃₃ = cofactor of a₃₃ \(= (−1)^{3+3} M_{33} = 1 \)

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(ii) The given determinant is \(\begin{vmatrix}1&0&4\\3&5&-1\\0&1&2\end{vmatrix}\)
By definition of minors and cofactors, we have:
M₁₁ = minor of a₁₁\(=\begin{vmatrix}5&-1\\1&2\end{vmatrix}\)=10+1=11
M₁₂ = minor of a₁₂\(=\begin{vmatrix}3&-1\\0&2\end{vmatrix}\)=6-0=6
M₁₃ = minor of a₁₃ \(=\begin{vmatrix}3&5\\0&1\end{vmatrix}\)=3-0=3
M₂₁= minor of a₂₁ \(=\begin{vmatrix}0&4\\1&2\end{vmatrix}\)=0-4=-4
M₂₂ = minor of a₂₂ =\(=\begin{vmatrix}1&4\\0&2\end{vmatrix}\)=2-0=2
M₂₃ = minor of a₂₃ \(=\begin{vmatrix}1&0\\0&1\end{vmatrix}\)=1-0=1
M₃₁ = minor of a₃₁\(=\begin{vmatrix}0&4\\5&-1\end{vmatrix}\)=0-20=-20
M₃₂ = minor of a₃₂ \(=\begin{vmatrix}1&4\\3&-1\end{vmatrix}\)=-1-12=-13
M₃₃ = minor of a₃₃ \(=\begin{vmatrix}1&0\\3&5\end{vmatrix}\)=5-0=5
A₁₁ = cofactor of a₁₁\(= (−1)^{1+1} M_{11} = 11\)
A₁₂ = cofactor of a₁₂ \(= (−1)^{1+2} M_{12} = −6\)
A₁₃ = cofactor of a₁₃ \(= (−1)^{1+3} M_{13} = 3\)
A₂₁ = cofactor of a₂₁ \(= (−1)^{2+1} M_{21} = 4\)
A₂₂ = cofactor of a₂₃ \(= (−1)^{2+2} M_{22} = 2\)
A₂₃ = cofactor of a₂₃ \(= (−1)^{2+3} M_{23} = −1\)
A₃₁ = cofactor of a₃₁ \(= (−1)^{3+1} M_{31} = −20\)
A₃₂ = cofactor of a₃₂ \(= (−1)^{3+2} M_{32} = 13\)
A₃₃ = cofactor of a₃₃ \(= (−1)^{3+3} M_{33} = 5\)