8
Feb 19

Determinants: starting simple

Determinants: starting simple

Previously  we looked at a motivating example to consider the determinant \det A=ad-bc of a 2\times 2 matrix

A=\left(\begin{array}{cc}a&b\\c&d\end{array}\right).

Using this basic example, now we formulate properties that uniquely define determinants of matrices of higher orders. The discussion is based mainly on Kurosh, Course in linear algebra, 9th edition, Moscow, 1968 (in Russian).

Observation 1. Homogeneity. If one of the rows of A is multiplied by a number k, then \det A gets multiplied by k,

\det \left(\begin{array}{cc}ka&kb\\c&d\end{array}\right)    =kad-kbc=k\det A, \det \left(\begin{array}{cc}a&b\\kc&kd\end{array}\right)=akd-bkc=k\det A.

Observation 2. Adding one of the rows of A to another doesn't change the value of the determinant:

\det \left(\begin{array}{cc}a+c&b+d\\c&d\end{array}\right)=(a+c)d-(b+d)c=\det A,

\det \left(\begin{array}{cc}a&b\\a+c&b+d\end{array}\right)=a(b+d)-b(a+c)=\det A.

To see the intuition behind these rules, recall that the purpose of the determinant is to verify whether the system Ax=y has solutions. Homogeneity means that if one of the equations of the system is multiplied by a nonzero constant, the solvability of the new system will be equivalent to the solvability of the original system. Similarly, if one of the equations of the system is added to another, solvability of the system as judged by the determinant will not change. This makes sense because multiplying a system equation by a nonzero constant or adding one equation to another does not change the information contained in the system.

Keep in mind an emerging general idea: the determinant discards any transformations of the matrix that do not impact its invertibility.

Observation 3. Determinant of the identity: \det I=1.

Taking these properties as axioms for matrices of higher order, we will show that they uniquely define determinants and develop a couple of rules involving determinants. One of them is the Leibniz formula for determinants and the other is Cramer's rule.

14
Jun 18

From invertibility to determinants: argument is more important than result

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 A be a 2\times 2 matrix. Using the condition AA^{-1}=I, find the invertibility condition and the inverse B=A^{-1}.

Solution. A good notation makes half of the solution. Denote

A=\left(\begin{array}{cc}a&b\\c&d\end{array}\right), B=\left(\begin{array}{cc}x&u\\y&v\end{array}\right).

It should be true that

\left(\begin{array}{cc}a&b\\c&d\end{array}\right)\left(\begin{array}{cc}x&u\\y&v\end{array}  \right)=\left(\begin{array}{cc}1&0\\0&1\end{array}\right).

This gives us four equations ax+by=1,\ au+bv=0,\ cx+dy=0,\ cu+dv=1. The notation guides us to consider two systems:

\left\{\begin{array}{c}ax+by=1\\cx+dy=0\end{array}\right. , \left\{\begin{array}{c}au+bv=0\\cu+dv=1\end{array}\right.

From the first system we have

\left\{\begin{array}{c}adx+bdy=d\\bcx+bdy=0\end{array}\right. .

Subtracting the second equation from the first we get (ad-bc)x=d. Hence, imposing the condition

(1) ad-bc\neq 0

we have x=\frac{d}{ad-bc}.

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:

y=\frac{-c}{ad-bc}, u=\frac{-b}{ad-bc}, v=\frac{a}{ad-bc}.

Thus (1) is the existence condition for the inverse and the inverse is

(2) A^{-1}=\frac{1}{ad-bc}\left(\begin{array}{cc}d&-b\\-c&a\end{array}\right).

Exercise 2. Check that (2) satisfies

(3) AA^{-1}=I.

Determinant

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 \det A=ad-bc is called a determinant of the matrix

A=\left(\begin{array}{cc}a&b\\c&d\end{array}\right).

The determinant of a general square matrix can be found using the Leibniz formula.

Exercise 3. Check that \det(AB)=(\det A)(\det B) (multiplicativity). Hint. Find the left and right sides and compare them. Here is the proof in the general case.

Exercise 4. How much is \det I?

Theorem. If the determinant of a square matrix A is different from zero, then that matrix is invertible.

The proof will be given later. Understanding functional properties of the inverse is more important than knowing the general expression for the inverse.

Exercise 5 (why do we need the determinant?) Prove that A is invertible if and only if \det A\neq 0.

Proof. Suppose \det A\neq 0. Then by the theorem above the inverse exists. Conversely, suppose the inverse exists. Then it satisfies (3). Apply \det to both sides of (3):

(4) (\det A)\det (A^{-1})=1.

This shows that \det A\neq 0.

Exercise 6 (determinant of an inverse) What is the relationship between \det A and \det A^{-1}?

Solution. From (4) we see that

(5) \det (A^{-1})=(\det A)^{-1}.

Exercise 7. For square matrices, existence of a right or left inverse implies existence of the other.

Proof. Suppose A,B are square and B is the right inverse:

(6) AB=I.

As in Exercise 5, this implies \det B\neq 0. By the theorem above we can use

(7) BB^{-1}=B^{-1}B=I.

By associativity (6) and (7) give BA=BA(BB^{-1})=B(AB)B^{-1}=BIB^{-1}=I.

The case of the left inverse is similar.