Determinant of a 2x2 matrix:

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:

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:

## Cramer’s Rule

Given a system of equations \(A \mathbf{x}=\mathbf{b}\), where \(A\) is a square matrix, we have the solution:

where \(A_{i}\) is the matrix formed by replacing the i-th column of \(A\) by the column vector \(\mathbf{b}\).