Determinants
\(\normalsize A=
\begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{bmatrix}
\)
\(|A|=\begin{vmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{vmatrix}\)
If \(A\) is a \(2\times 2\) matrix, then the determinant is defined as:
\(|A|=a_{11}a_{22}-a_{21}a_{12}\)
where the result is a scalar.
For any \(n\times n\) matrix where \(n\ge 3\), let \(A=(a_{ij})\) be the \(n\times n\) matrix and \(a_{kp}\) be an arbitrary
entry in that matrix.
1. The minor of \(a_{kp}\) is the determinant of the matrix remaining after deleting row \(k\) and column \(p\)
of matrix \(A\), and is denoted as \(MIN(a_{kp})\).
2. The cofactor of \(a_{kp}\) is the number \((-1)^{k+p}MIN(a_{kp})\), and is denoted as \(COF(a_{kp})\).
The determinant of an \(n\times n\) matrix \(A=(a_{ij})\) is the number:
\(a_{11}COF(a_{11})+a_{21}COF(a_{21})+\)...\(+a_{n1}COF(a_{n1})\).
For example, the determinant of a \(3\times 3\) matrix is:
\(|A|=\begin{vmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{vmatrix}=\)
\(a_{11}\times\begin{vmatrix}
a_{22} & a_{23}\\
a_{32} & a_{33}
\end{vmatrix}\)
\(-a_{12}\times\begin{vmatrix}
a_{21} & a_{23}\\
a_{31} & a_{33}
\end{vmatrix}\)
\(+a_{13}\times\begin{vmatrix}
a_{21} & a_{22}\\
a_{31} & a_{32}
\end{vmatrix}\)
\(=a_{11}\times(a_{22}a_{33}-a_{23}a_{32})\)
\(-a_{12}\times(a_{21}a_{33}-a_{23}a_{31})\)
\(+a_{13}\times(a_{21}a_{32}-a_{22}a_{31})\)
Notes:
1. Generally, the cofactors from the first column are used, but the cofactors from any row or column can be used with the same results.
2. Interchanging any two rows or columns of the matrix does not alter the value of the determinant, but does change its sign.
3. Any matrix with either a row or column of all zeros has a determinant of zero.
4. Any matrix with any two rows or columns with equal or proportional values has a determinant of zero.
The following algebraic properties are true in general:
\(|AB|=|A||B|\)
\(|AB|=|BA|\)
Also, if \(T\) is the transpose of \(A\), then:
\(|A|=|T|\)
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