The determinant of a matrix

Saturday, 9 June 2018

4:15 PM

The determinant of a matrix can be used to find out if a matrix is invertible

Notation:  or

 

Conditions for a determinant to exist (and be non-zero)

• Matrix is a square of size

 

Consequences of a determinant

If , then matrix A has an inverse

If , then matrix A does not have an inverse

 

Determinant of a  matrix

For

 

Determinant of an  matrix

• Work along the first row
• 

• Flip the sign for each operation

 

 

The minor of a matrix

The matrix A of minor , is the matrix A without row  or column

Notation:

 

Other properties

• 
• If rows (or columns) are interchanged, the determinant changes sign for each interchange
• If a matrix contains a zero row/column, then the determinant is zero
•     
• If a row (or column) is a multiple of another row (or column) the determinant is zero
• 
• If  is a square matrix in row-echelon form, then the determinant is the product of the diagonal