To introduce determinants of square matrices
To learn the formula for the determinant of 2x2 matrices
To understand how to visualize determinants
To use Cramer's Rule to solve matrix equations
In this packet we are introduced to determinants of 2x2 matrices. We learn a bit about what determinants mean, and how we can use them.
Is there an easy way to tell whether or not a matrix is invertible? As it turns out there is. Every square matrix is associated with a number, called the determinant of the matrix, which can be used to determine whether or not a matrix has an inverse. If a matrix has a non-zero determinant, then it is invertible; if the determinant equals zero, then the matrix does not have an inverse.
For a 2x2 matrix
the determinant is defined to be the value (ad-bc),
and is often denoted .
If we have a matrix M, we will often write |M| or det(M) to mean the determinant of M.
Property 1: det(M-1) = 1/det(M)
The determinant of the inverse of M is the reciprocal of the determinant of M.
Property 2: M-1 = 1/det(M) * adj(M)
The inverse of the matrix M is the reciprocal of the determinant times the adjugate of M. The notation adj(M) stands for the adjugate matrix of M.
If M = , then adj(M) =
Using the previous example, we have:
Knowing the determinant of a matrix can help us more easily solve matrix equations without using Gaussian elimination. Instead, we can use Cramer's Rule:
If A is a square matrix such that the determinant |A| is non-zero, then the equation Ax=b has a unique solution x, whose entries xi=|Ai|/|A|, where Ai is the matrix obtained by replacing the ith column of A by the vector b.
Depending on the matrix, using Cramer's rule can greatly simplify the task of finding solutions to linear systems. The rule tends to work best whenever determinants are easy to calculate, such as when the matrix is small, or when it contains many zero entries.
Up next is a video example in which we apply Cramer's rule to solve a matrix equation.
We work through an example using Cramer's rule to solve matrix equation.
Understanding how to find determinants is one thing, but what do determinants actually mean? For 2x2 matrices, we can understand the determinant as the area of a parallelogram in the xy-plane. The vertices of this parallelogram are given by the matrix columns.
It turns out that this same analogy works for larger matrices - the determinant of a 3x3 matrix, for instance, corresponds to the volume of a parallelepiped in three dimensional space.