Determinant of a square matrix

• The determinant of a matrix is a scalar value that encodes information about the linear transformation represented by the matrix.

• The determinant of a square matrix can be calculated using various methods, including cofactor expansion, row or column operations, and LU decomposition.
• If a matrix has a row or column of zeros, then its determinant is zero.
• The determinant of a diagonal matrix is the product of its diagonal elements.
• The determinant of a product of matrices is equal to the product of their determinants, i.e., det(AB) = det(A) det(B). However, det(A+B) is not equal to det(A) + det(B) in general.
• The determinant of the transpose of a matrix is equal to the determinant of the original matrix, i.e., det(A) = det(A^T).
• A matrix is invertible if and only if its determinant is nonzero. In other words, a matrix is singular if and only if its determinant is zero.
• The absolute value of the determinant of a matrix represents the factor by which the matrix scales the area or volume of a geometric object, depending on the dimensionality of the matrix.

 

Important Formulas to remember

Determinant of a Square Matrix: $$|A| = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij}$$