Determinants

Definition: The determinant is a special number that can be calculated from a matrix. For a 2x2 matrix the determinant may be calculated as ad-bc, where a, b, c and d are the elements of the matrix.
A matrix that does not have a determinant is known as a singular matrix, while those that do have are referred to as non-singular.
The determinant of a 3x3 matrix can be calculated using the rule of Sarrus. Firstly, write down the elements of the first two columns to the right of the matrix. Secondly, add the products of the diagonals going from top left, and subtract the products of the diagonals going from top right.
The determinant of an identity matrix of any size is always 1.
The determinant of a diagonal or triangular matrix is simply the product of the entries on the main diagonal.
If a matrix and its transpose are both square matrices (i.e., they have the same number of rows and columns), then they have the same determinant. Transposition does not affect determinant value.
If you multiply one row or column of a matrix by a scalar, the determinant is also multiplied by that scalar.
If two rows or columns in a matrix are swapped, the sign of the determinant changes.
If two rows or columns of a matrix are identical, the determinant of that matrix is zero.
The determinant of the product of two square matrices equals the product of their determinants. Essentially, det(AB) = det(A) * det(B) is true for square matrices A and B.
Cramer’s Rule: In a system of linear equations, if the determinant of the coefficient matrix is not zero, a unique solution exists. This is given by the ratio of the determinant formed by replacing a column with the constant terms, to the determinant of coefficient matrix itself.
An important use of determinants is in finding the eigenvalues of a matrix. For a matrix to have an eigenvector solution, it is required that the determinant of the matrix subtracted from a scalar times the identity matrix be equal to zero.
Determinants are valuable tools throughout all areas of linear algebra, from evaluating the inverse of a matrix to solving system of linear equations. Practicing calculating determinants by hand is useful for developing comfort and efficiency with this important mathematical concept.