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Mathematics Question on Determinants

Write Minors and Cofactors of the elements of following determinants:
I. ∣100\010\001∣\begin{vmatrix}1&0&0\\\0&1&0\\\0&0&1\end{vmatrix}II. ∣104\35−1\012∣\begin{vmatrix}1&0&4\\\3&5&-1\\\0&1&2\end{vmatrix}

Answer

I. The given determinant is ∣100\010\001∣\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=∣10\01∣\begin{vmatrix}1&0\\\0&1\end{vmatrix}=1

M12 = minor of a12=∣00\01∣\begin{vmatrix}0&0\\\0&1\end{vmatrix}=0

M13 = minor of a13 =∣01\00∣\begin{vmatrix}0&1\\\0&0\end{vmatrix}=0

M21 = minor of a21 =∣00\01∣\begin{vmatrix}0&0\\\0&1\end{vmatrix}=0

M22 = minor of a22 =∣10\01∣\begin{vmatrix}1&0\\\0&1\end{vmatrix}=1

M23 = minor of a23 =∣10\00∣\begin{vmatrix}1&0\\\0&0\end{vmatrix}=0

M31 = minor of a31=∣00\10∣\begin{vmatrix}0&0\\\1&0\end{vmatrix}=0

M32 = minor of a32 =∣10\00∣\begin{vmatrix}1&0\\\0&0\end{vmatrix}=0

M33 = minor of a33 =∣10\01∣\begin{vmatrix}1&0\\\0&1\end{vmatrix}=1

A11 = cofactor of a11=(−1)1+1M11=1= (−1)^{1+1} M_{11} = 1
A12 = cofactor of a12 =(−1)1+2M12=0= (−1)^{1+2} M_{12} = 0
A13 = cofactor of a13 =(−1)1+3M13=0= (−1)^{1+3} M_{13} = 0
A21 = cofactor of a21 =(−1)2+1M21=0= (−1)^{2+1} M_{21} = 0
A22 = cofactor of a22 =(−1)2+2M22=1= (−1)^{2+2} M_{22} = 1
A23 = cofactor of a23 =(−1)2+3M23=0= (−1)^{2+3} M_{23} = 0
A31 = cofactor of a31 =(−1)3+1M31=0= (−1)^{3+1} M_{31} = 0
A32 = cofactor of a32 =(−1)3+2M32=0= (−1)^{3+2} M_{32} = 0
A33 = cofactor of a33 =(−1)3+3M33=1= (−1)^{3+3} M_{33} = 1

(ii) The given determinant is ∣104\35−1\012∣\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=∣5−1\12∣=\begin{vmatrix}5&-1\\\1&2\end{vmatrix}=10+1=11
M12 = minor of a12=∣3−1\02∣=\begin{vmatrix}3&-1\\\0&2\end{vmatrix}=6-0=6
M13 = minor of a13 =∣35\01∣=\begin{vmatrix}3&5\\\0&1\end{vmatrix}=3-0=3
M21= minor of a21 =∣04\12∣=\begin{vmatrix}0&4\\\1&2\end{vmatrix}=0-4=-4
M22 = minor of a22 ==∣14\02∣=\begin{vmatrix}1&4\\\0&2\end{vmatrix}=2-0=2
M23 = minor of a23 =∣10\01∣=\begin{vmatrix}1&0\\\0&1\end{vmatrix}=1-0=1
M31 = minor of a31=∣04\5−1∣=\begin{vmatrix}0&4\\\5&-1\end{vmatrix}=0-20=-20
M32 = minor of a32 =∣14\3−1∣=\begin{vmatrix}1&4\\\3&-1\end{vmatrix}=-1-12=-13
M33 = minor of a33 =∣10\35∣=\begin{vmatrix}1&0\\\3&5\end{vmatrix}=5-0=5
A11 = cofactor of a11=(−1)1+1M11=11= (−1)^{1+1} M_{11} = 11
A12 = cofactor of a12 =(−1)1+2M12=−6= (−1)^{1+2} M_{12} = −6
A13 = cofactor of a13 =(−1)1+3M13=3= (−1)^{1+3} M_{13} = 3
A21 = cofactor of a21 =(−1)2+1M21=4= (−1)^{2+1} M_{21} = 4
A22 = cofactor of a23 =(−1)2+2M22=2= (−1)^{2+2} M_{22} = 2
A23 = cofactor of a23 =(−1)2+3M23=−1= (−1)^{2+3} M_{23} = −1
A31 = cofactor of a31 =(−1)3+1M31=−20= (−1)^{3+1} M_{31} = −20
A32 = cofactor of a32 =(−1)3+2M32=13= (−1)^{3+2} M_{32} = 13
A33 = cofactor of a33 =(−1)3+3M33=5= (−1)^{3+3} M_{33} = 5