Question #197274

Comupute all the minors and the cofactors of: [1 2 3]

[2 0 1]

[2 3 4]


1
Expert's answer
2021-05-24T15:29:55-0400

Solution.


A=(123201234)A=\begin{pmatrix} 1& 2&3\\ 2& 0&1\\ 2&3&4 \end{pmatrix}

The elements of the minor of the matrix A will be given by

M1,1=0134=3M_{1,1}=\begin{vmatrix} 0 & 1\\ 3& 4 \end{vmatrix}=-3

M1,2=2124=6M_{1,2}=\begin{vmatrix} 2& 1\\ 2& 4 \end{vmatrix}=6

M1,3=2023=6M_{1,3}=\begin{vmatrix} 2 & 0\\ 2& 3 \end{vmatrix}=6

M2,1=2334=1M_{2,1}=\begin{vmatrix} 2& 3\\ 3& 4 \end{vmatrix}=-1

M2,2=1324=2M_{2,2}=\begin{vmatrix} 1 & 3\\ 2& 4 \end{vmatrix}=-2

M2,3=1223=1M_{2,3}=\begin{vmatrix} 1 & 2\\ 2& 3 \end{vmatrix}=-1

M3,1=2301=2M_{3,1}=\begin{vmatrix} 2 & 3\\ 0& 1 \end{vmatrix}=2

M3,2=1321=5M_{3,2}=\begin{vmatrix} 1& 3\\ 2& 1 \end{vmatrix}=-5

M3,3=1220=4M_{3,3}=\begin{vmatrix} 1& 2\\ 2& 0 \end{vmatrix}=-4

Hence, M=(366121254)M=\begin{pmatrix} -3& 6&6 \\ -1& -2&-1\\ 2&-5&-4 \end{pmatrix}

The cofactor matrix will be given by Ci,j=(1)i+jMi,j.C_{i,j}=(-1)^{i+j}M_{i,j}.

C=(366121254)C=\begin{pmatrix} -3& -6&6 \\ 1& -2&1\\ 2&5&-4 \end{pmatrix}


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