1.4 Matrix Algebra · Linear Algebra Refresher

Matrix algebra looks like ordinary algebra with two loud exceptions: multiplication doesn't commute, and you can't cancel. Almost every true/false trap in this chapter lives in one of those two gaps.

Theorem 1.4.1 · Algebraic rules

For all scalars

α, β

and matrices of compatible size:

A + B = B + A

;

(A + B) + C = A + (B + C)

;

(A B) C = A (B C)

;

A (B + C) = A B + A C

;

(A + B) C = A C + B C

;

α (A B) = (α A) B = A (α B)

; plus the usual scalar rules.

Associativity is why you can write $A B C$ without parentheses, and it's the step that makes $A^{k}$ well defined.

!The three that fail

A B \neq = B A

A B = O

does not force

A = O

B = O

A B = A C

with

A \neq = O

does not force

B = C

. Cancellation needs

A

to be nonsingular, not merely nonzero.

The identity and powers

Definition · Identity matrix

I = (δ_{ij})

with

δ_{ij} = 1

i = j

and

0

otherwise. Then

I A = A I = A

, and

I = (e_{1}, e_{2}, \dots, e_{n})

— its columns are the standard basis vectors.

Worked example

Powers of a rank-one matrix

0/3 steps

Find

A^{2}

A^{3}

, and a formula for

A^{n}

A = (1111)

Inverses

Definition · Nonsingular / invertible

n \times n

matrix

A

is nonsingular if there is a matrix

B

with

A B = B A = I

. Then

B

is unique, denoted

A^{- 1}

. If no such

B

exists,

A

is singular.

Uniqueness is a one-liner: if $B$ and $C$ both work, $B = B I = B (A C) = (B A) C = I C = C$ . And note that 'singular/nonsingular' applies only to square matrices — never call a $3 \times 5$ matrix singular.

Theorem 1.4.2

A

and

B

are nonsingular

n \times n

matrices, then

A B

is nonsingular and

(A B)^{- 1} = B^{- 1} A^{- 1}

By induction, $(A_{1} A_{2} \dots A_{k})^{- 1} = A_{k}^{- 1} \dots A_{2}^{- 1} A_{1}^{- 1}$ . Think of putting on socks then shoes: to undo, remove shoes first.

$(A^{T})^{T} = A$
$(α A)^{T} = α A^{T}$
$(A + B)^{T} = A^{T} + B^{T}$
$(A B)^{T} = B^{T} A^{T}$ — same order reversal as the inverse rule

Check yourself

A

and

B

are nonsingular

n \times n

matrices, then

(A B)^{- 1}

equals:

T / F

A B = O

for

n \times n

matrices, then

A = O

B = O

T / F

A B = A C

and

A \neq = O

, then

B = C

1.4Matrix Algebra

What you must be able to do

The identity and powers

Powers of a rank-one matrix

Inverses

Check yourself