1.5 Elementary Matrices · Linear Algebra Refresher

Row operations aren't just moves on paper — each one is left multiplication by an invertible matrix. That reframing gives you the equivalence theorem for nonsingularity and a clean algorithm for

A^{- 1}

Applying a sequence of elementary matrices $E_{1}, \dots, E_{k}$ to both sides of $A x = b$ turns it into $U x = c$ with $U = E_{k} \dots E_{1} A$ and $c = E_{k} \dots E_{1} b$ . Since every $E_{i}$ is invertible, the two systems are equivalent.

Type	Operation on $I$	$E A$ does	$A E$ does
I	Swap two rows	Swaps those rows of $A$	Swaps the corresponding columns
II	Scale a row by $α \neq = 0$	Scales that row of $A$	Scales that column
III	Add $α \times$ (row $j$ ) to row $i$	Same row operation on $A$	Adds $α \times$ (col $i$ ) to col $j$

Left multiply ⟹ row operation. Right multiply ⟹ column operation. Mixing these up is a classic exam slip.

Theorem 1.5.1

E

is elementary, then

E

is nonsingular and

E^{- 1}

is an elementary matrix of the same type.

Obvious once you think of undoing the move: swap back, scale by $1/ α$ , subtract what you added.

Definition · Row equivalence

B

is row equivalent to

A

B = E_{k} \dots E_{1} A

for some elementary matrices — i.e.

B

is reachable from

A

by finitely many row operations. The relation is symmetric and transitive.

The nonsingularity theorem

Theorem 1.5.2 · Equivalent conditions

For an

n \times n

matrix

A

, the following are equivalent: (a)

A

is nonsingular; (b)

A x = 0

has only the trivial solution; (c)

A

is row equivalent to

I

This list keeps growing all term — det ≠ 0 (§2.2), columns independent / a basis (§3.6), rank $n$ , no eigenvalue equal to 0 (§6.1). Learn it as one object.

Corollary 1.5.3

A x = b

(

n

equations,

n

unknowns) has a unique solution iff

A

is nonsingular.

Computing $A^{- 1}$

If $A$ is nonsingular there are elementary matrices with $E_{k} \dots E_{1} A = I$ . Multiply on the right by $A^{- 1}$ : $E_{k} \dots E_{1} I = A^{- 1}$ . So the same sequence of row operations that turns $A$ into $I$ turns $I$ into $A^{- 1}$ — run them side by side.

rref (A ∣ I) = (I ∣ A^{- 1})

Worked example

Inverting a $3 \times 3$ matrix, then using it

0/5 steps

Compute

A^{- 1}

and use it to solve

A x = b

with

b = (12, - 12, 8)^{T}

A = 1 - 1 2 4 - 2 2 303

!Exam trap

Computing

A^{- 1}

to solve one system is wasteful — plain elimination on

(A ∣ b)

is faster and less error-prone. Invert only when you need

A^{- 1}

itself or must solve many systems with the same

A

. If the left half fails to reach

I

A

is singular; stop.

Diagonal and triangular matrices

Upper triangular: $a_{ij} = 0$ for $i > j$ . Lower triangular: $a_{ij} = 0$ for $i < j$ .
Diagonal: $a_{ij} = 0$ whenever $i \neq = j$ — both upper and lower triangular at once.
A triangular matrix may have zeros on the diagonal, in which case it is singular. Strict triangular form for a system additionally requires a nonzero diagonal.

Check your work

Determinant & inverse

Reduces to triangular form, tracking every sign flip. Square matrices only.

Square matrix

2	1	3
4	2	1
6	-3	4

det =-60

status

nonsingular

Ax = b

unique solution

A⁻¹

-11/60	13/60	1/12
1/6	1/6	-1/6
2/5	-1/5	0

Invert a matrix and watch the elimination that produces it.

Check yourself

You multiply a

3 \times 3

matrix

A

on the right by the elementary matrix formed by swapping rows 1 and 2 of

I

. The result is:

T / F

n \times n

matrix

A

is nonsingular if and only if

A

is row equivalent to

I

T / F

Every upper triangular matrix is nonsingular.

1.5Elementary Matrices

What you must be able to do

The nonsingularity theorem

Computing $A^{- 1}$

Inverting a $3 \times 3$ matrix, then using it

Diagonal and triangular matrices

Determinant & inverse

Check yourself

1.5Elementary Matrices

What you must be able to do

The nonsingularity theorem

Computing A−1

Inverting a 3×3 matrix, then using it

Diagonal and triangular matrices

Determinant & inverse

Check yourself

Computing $A^{- 1}$

Inverting a $3 \times 3$ matrix, then using it