Determinants are extremely useful in solving linear equations - we also encounter determinants in the expression of the cross product of two vectors. Imagine solving this set of simultaneous linear equations:

Which can also be written in matrix/vector form:

or more compactly as:

where it is understood that A is a matrix and x and b are vectors. In this example, A is a 3x3 matrix and the vectors have three components. You can also imagine a system of only two equations where the matrix A would be a 2x2 matrix and the vectors would have two components.

Determinant of a Matrix

Start with a 2x2 matrix

Here is how we define the determinant:

Now go to a 3x3 matrix:

and you already know how to expand the 2x2 determinant. This method of writing the determinant of a 3x3 matrix in terms of 2x2 determinants can be generalized to find the determinant of higher dimensional matrices.

Examples

You can verify:

Use in Solving Simultaneous Linear Equations

Going back to the 3 simultaneous equations at the top of this note:

Let us assume that the x's are unknown and the other numbers (a's and b's) are all given. We form the determinant of the coefficients:

As long as the equations are linearly independent - this determinant will not vanish. In this case, the solutions are:

As an exercise show that the solutions to:

are: