21
Jun 18

## Matrix algebra: questions for repetition

Matrix algebra is a bunch of dreadful rules is all that many students remember after studying it. It's a relief to know that a simple property like $AA^{-1}=I$ is more important than remembering how to calculate an inverse. The theoretical formula for the inverse for the most part of this course can be avoided; if you need to find the inverse of a numerical matrix, you can use Excel.

### First things first

Three big No's: 1) there is no commutativity, in general, 2) determinants don't apply to non-square matrices, and 3) don't try to invert a non-square matrix. There are ways around these problems but all of them are deficient, so better stick to good cases.

Three big ideas: 1) the analogy with real numbers is the best guide to study matrices, 2) the matrix product definition is motivated by the desire to compactify a system of equations, 3) symmetric matrices have properties closest to those of real numbers.

Three surprises: 1) in general, matrices don't commute (can you give an example?), 2) a nonzero matrix is not necessarily invertible (can you give an example?), 3) when you invert a product, you have to change the order of the factors (same goes for transposition). These two properties are called reverse order laws.

Comforting news: 1) properties of summation of numbers have complete analogs for matrices, 2) in case of multiplication, it's good to know that existence of a unity, associativity and distributivity generalize to matrices.

### Particulars and extensions

Answer the following questions, with proofs where possible. None of the answers requires long boring calculations.

Multiplication. 1) If $A^{2}$ exists, what can you say about $A$? 2) If the last row of $A$ is zero and the product $AB$ exists, what can you say about this product? 3) Where did we use associativity of multiplication?

In what way the rules for the inverse of a product and transposed of a product are similar? Can you tell any differences between them?

Commutativity: 1) If two matrices commute, do you think their inverses commute? 2) Does a matrix commute with its inverse?

Properties of inverses: 1) inverse of an inverse, 2) inverse of a product, 3) inverse of a transpose.

Properties of determinants: 1) why we need them, 2) determinant of a product, 3) determinant of an inverse, 4) determinant of a transpose. 5) Prove the multiplication rule for the determinant of the product of three matrices.

Properties of the identity matrix: 1) use the definition of the inverse to find the inverse of the identity matrix, 2) do you think the identity matrix commutes with any other matrix? 3) Can you name any matrices, other than the identity, satisfying the equation $A^{2}=A?$ If a matrix satisfies this equation, what can you say about its determinant? 4) What is the determinant of the identity matrix?

If a nonzero number is close to zero, then its inverse must be a large number (in absolute value). True or wrong? Can you indicate any analogs of this statement for matrices?

Suppose matrices $A,B$ are given and $\det A\neq 0.$ How would you solve the linear matrix equation $AX=B$ for $X?$

Symmetric matrices: 1) For any matrix $A,$ both matrices $AA^T$ and $A^TA$ are symmetric. True or wrong? 2) If a matrix is symmetric and its inverse exists, will the inverse be symmetric?

14
Jun 18

## 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.