From invertibility to determinants: argument is more important than result
Interestingly enough, determinants appeared before matrices.
Invertibility condition and expression for the inverse
Exercise 1. Let be a matrix. Using the condition find the invertibility condition and the inverse
Solution. A good notation is half of the solution. Denote
It should be true that
This gives us four equations The notation guides us to consider two systems:
From the first system we have
Subtracting the second equation from the first we get Hence, imposing the condition
Definition. The method for solving a system of linear equations applied here is called an elimination method: we multiply the two equations by something in such a way that after subtracting one equation from another one variable is eliminated. There is also a substitution method: you solve one equation for one variable and plug the resulting expression into another equation. The elimination method is better because it allows one to see the common structure of the resulting expressions.
Use this method to find the other variables:
Thus (1) is the condition for the existence of the inverse and the inverse is
Exercise 2. Check that (2) satisfies
The problem with determinants is that they are needed early in the course but their theory requires developed algebraic thinking. I decided to stay at the intuitive level for a while and delay the theory until Section 8.
Definition. The expression is called a determinant of the matrix The determinant of a general square matrix can be found using the Leibniz formula.
Exercise 3. Check that (multiplicativity). Hint. Find the left and right sides and compare them. Here is the proof in the general case.
Exercise 4. How much is
Theorem. If the determinant of a square matrix is different from zero, then that matrix is invertible.
The proof will be given later. Note that we don't need the general expression for the inverse.
Exercise 5 (why do we need the determinant?) Prove that is invertible if and only if
Proof. Suppose Then by the theorem above the inverse exists. Conversely, suppose the inverse exists. Then it satisfies (3). Apply to both sides of (3):
This shows that
Exercise 6 (determinant of an inverse) What is the relationship between and
Solution. From (4) we see that
Exercise 7. For square matrices, existence of a right or left inverse implies existence of the other.
Proof. Suppose are square and is the right inverse:
As in Exercise 5, this implies . By the theorem above we can use
By associativity (6) and (7) give
The case of the left inverse is similar.