(i) Assume ${{M}_{ij}}$ and ${{C}_{ij}}$ represents the minor and co–factor of an element, where i and j represent the row and column. The minor of the matrix can be obtained for a particular element by removing the row and column where the element is present. Then finding the absolute value of the matrix newly formed.
And, ${{C}_{ij}}={{\left( -1 \right)}^{i+j}}\times {{M}_{ij}}$
It is given in the question that,
$A=\left[ \begin{matrix}
5 & 20 \\
0 & -1 \\
\end{matrix} \right]$
From the given information about the matrix in the question we get,
${{M}_{11}}=-1$
${{M}_{21}}=20$
${{C}_{11}}={{\left( -1 \right)}^{1+1}}\times {{M}_{11}}$
$=1\times -1$
$=-1$
${{C}_{21}}={{\left( -1 \right)}^{2+1}}\times {{M}_{21}}$
$=20\times -1$
$=-20$
Then expanding along the first column we have
$\left| A \right|={{a}_{11}}\times {{C}_{11}}+{{a}_{21}}\times {{C}_{21}}$
$=5\times \left( -1 \right)+0\times \left( -20 \right)$
$=-5$
(ii) Assume ${{M}_{ij}}$and ${{C}_{ij}}$ represents the minor and co–factor of an element, where i and j represent the row and column. The minor of matrix can be obtained for particular element by removing the row and column where the element is present. Then finding the absolute value of the matrix newly formed.
Then, ${{C}_{ij}}={{\left( -1 \right)}^{i+j}}\times {{M}_{ij}}$
It is given in the question,
$A=\left[ \begin{matrix}
-1 & 4 \\
2 & 3 \\
\end{matrix} \right]$
From the given matrix in the question we get
${{M}_{11}}=3$
${{M}_{21}}=3$
${{C}_{11}}={{\left( -1 \right)}^{1+1}}\times {{M}_{11}}$
$=1\times 3$
$=3$
${{C}_{21}}={{\left( -1 \right)}^{2+1}}\times 4$
$=-1\times 4$
$=-4$
Then expanding along the first column we have
$\left| A \right|={{a}_{11}}\times {{C}_{11}}+{{a}_{21}}\times {{C}_{21}}$
$=-1\times 3+2\times \left( -4 \right)$
$=-11$