The determinant of a matrix is a single scalar value which is used when calculating the inverse of a square matrix or when solving systems of linear equations. The matrix determinant is usually represented by , or where is the matrix.

The matrix determinant for a matrix can be defined as follows

The matrix determinant for a matrix can be then be found using the Laplace expansion

Geometric Intuition

The matrix determinant calculates the area (or volume in higher dimensions) of the "parallelogram" formed between the column vectors of the matrix. For example given the following matrix

The determinant would be the area formed between the vectors and . In higher dimensions the same is true but instead it's the volume of the higher dimensional equivalent of a parallelogram.

Remark

When the column vectors in the matrix are all in the same plane the determinant will be 0.

Identities

Where , , and are 3 dimensional vectors.

Examples

Area of a Triangle

Find the area of the triangle with vertices , , and .

Taking the absolute value we get an area of

Volume of a Higher Dimensional Parallelogram

Find the volume of the parallelepiped formed by the vectors , , and .

Properties

• The determinant of the identity matrix is 1.
• The determinant of a triangle matrix is the product of it's diagonals elements.
• If a matrix is invertable
• For any square matrix ,
• For any two square matrices and ,

Row operations on a matrix have the following effects on it's determinant

• Swapping two rows of changes the determinant by a factor of .
• Scaling a row by multiplies the determinant by .
• Row addition has no effect and leaves the determinant unchanged.