A matrix determinant requires a few more steps. It is computed by continuously breaking matrices down into smaller matrices until the 2x2 form is reached in a process called Expansion by Minors also known as Cofactor Expansion. We first define the minor matrix of as the matrix which is derived from by eliminating the row and column. For example, the 3x3 matrix and its minor (given by eliminating row 1 and column 1) are shown below:
The determinant of is given by the following expansion by the first row of :
The second term is negative in this expansion because the sign of the cofactors is given by . Taking these definitions together we can write the determinant by expansion along row formally as follows:
It is valid to expand about any row or column. This can be especially useful if expansion about a particular row or column results in value for one or more of the cofactors since this eliminates an entire minor matrix from the calculation.
Excercise 3-7.
Find the following matrix determinant: