Determinant of a 2x2 matrix:
\begin{equation*}
\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=ad-bc
\end{equation*}
The minor of an element \(a_{ij}\), denoted \(M_{ij}\) is the determinant of the matrix where row \(i\) and column \(j\) have been deleted.
The cofactor of an element \(a_{ij}\) is \(\left(-1)^{i+j}M_{ij}\).
The determinant of a matrix is the sum of the product of each element in a row/column with its cofactor. For example, using the j-th column:
\begin{equation*}
\sum_{i=1}^{n}a_{ij}A_{ij}
\end{equation*}
Some properties of determinants:
- If you interchange two rows or columns, the sign of the determinant is reversed.
- If two rows or two columns are identical, the determinant is 0.
- If you multiply a row/column by \(k\), then the determinant is \(k|M|\).
- If to each entry of a row (or a column) of a square matrix \(A\) is added a non-zero multiple of the corresponding entry of another row (or column), then the determinant of the resulting matrix is \(|A|\) (i.e. unchanged).
The adjoint of a matrix \(A\) is the matrix with all elements replaced by its cofactors.
The inverse of a matrix is:
\begin{equation*}
A^{-1}=\frac{1}{|A|} \operatorname{adj} A
\end{equation*}
Cramer’s Rule
Given a system of equations \(A \mathbf{x}=\mathbf{b}\), where \(A\) is a square matrix, we have the solution:
\begin{equation*}
x_{i}=\frac{|A_{i}|}{|A|} \quad i=1, \ldots, n
\end{equation*}
where \(A_{i}\) is the matrix formed by replacing the i-th column of \(A\) by the column vector \(\mathbf{b}\).