A matrix’s determinant is the scalar value or number calculated using a square matrix. The square matrix could be 2 x 2, 3 x 3, 4 x 4, or any other type where the number of columns and rows are equal, such as n x n.

A matrix’s determinant is represented by two vertical lines or simply by writing det and the matrix name. For example, |A|, det(A), det A.

Properties of Determinants

If the rows of a determinant are changed into columns and vice verse, the value of the determinant remains unchanged. i.e. |A| = |A^{t}|

If any two rows (or columns) are interchanged, the value of the resulting determinant is additive inverse of the value of the original determinant. i.e. |A| = -|A|

If any two rows or columns of the determinant are identical, the determinant’s value is zero.

If the elements of a row (or column) of a determinant are added (subtracted) k-times the corresponding elements of another row (column), the value of the determinant so obtained remains unchanged.

If the elements of a row (column) of a matrix are multiplied by the same number , say k, the determinant of the matrix so obtained is k-times the determinant of the original matrix.

If the elements of any row or column of determinant are sum (difference) of two or more elements, then the determinant can be expressed as sum (difference) of two or more determinants.