Square of a determinant of matrix A=\[\begin{vmatrix}a&b\\c&d\end{vmatrix}\]
=\[ (ad-bc)^2\]
Multiplication of determinant by a scalar- If a determinant is multiplied by a scalar then it is important to note that any one row or coulumn of the determinant is multiplied by the scalar not all the rows or columns unlike the multiplicatio by scalar to a given matrix.
Example If determinant A= \[ \begin{vmatrix}2&3&1\\1&2&5\\1&2&1\end{vmatrix}\]
then 5A=\[ \begin{vmatrix}10&15&5\\1&2&5\\1&2&1\end{vmatrix}\] multiplication of scalar to first row
or 5A =\[ \begin{vmatrix}2&3&1\\5&10&25\\1&2&1\end{vmatrix}\] multiplication of scalar to 2nd row
or 5A =\[ \begin{vmatrix}2&3&1\\1&12&5\\5&10&5\end{vmatrix}\]multiplication of scalar to 3rd row
It is to note that all the three cases will give same value of determinant.If all these three operation is applied on three columns then also we get same value of determinant.
Determinant of Identity matrix
Determinant of identity matrix is always 1
Multiplication of Identity matix by a scalar
If A is Identity matrix then determinant of 5A= |5A|=5
If A is Identity matrix then determinant of 10A= |10A|=10
