Systems of linear equations — matrix form Ax = b: Question bank

Notation reminder used throughout (all defined in the parent):

$A$ is the $m \times n$ coefficient matrix — $m$ rows (equations), $n$ columns (unknowns).
$x$ is the vector of unknowns, $b$ the right-hand side.
$[A ∣ b]$ is the augmented matrix: $A$ with the column $b$ glued on the right.
$rank$ counts independent rows/columns; $Col (A)$ is the span of the columns; the null space is all $x$ with $A x = 0$ .

True or false — justify

TF1. "If

A

m \times n

with

m > n

(more equations than unknowns), the system has no solution."

False — solvability depends on rank, not row count; if the extra rows are dependent,

b

can still lie in

Col (A)

and a solution exists. See Rank of a matrix.

TF2. "

A x = 0

can be inconsistent (have no solution)."

False —

x = 0

always satisfies

A x = 0

, so a homogeneous system is never inconsistent; the only question is whether nonzero solutions also exist.

TF3. "If

det A = 0

then

A x = b

has no solution."

False —

det A = 0

only kills the unique solution; the system may still have infinitely many solutions (if

b \in Col (A)

) or none (if it doesn't). See Determinant.

TF4. "If

A x = b

has two distinct solutions, it has infinitely many."

True — if

A x_{1} = A x_{2} = b

then

A (x_{1} - x_{2}) = 0

, so a nonzero null-space vector exists, and

x_{1} + t (x_{1} - x_{2})

solves it for every real

t

TF5. "For a square

A

A x = b

has a unique solution for every

b

iff

A

is invertible."

True — invertibility means the columns span all of

R^{n}

and are independent, so every

b

is hit exactly once. See Inverse of a matrix.

TF6. "Adding a redundant equation (a copy of an existing one) can change the solution set."

False — a duplicate row leaves both

rank (A)

and

rank ([A ∣ b])

unchanged, so the solution set is identical.

TF7. "If the columns of

A

are linearly independent, the solution to

A x = b

(when it exists) is unique."

True — independent columns mean the null space is only

{0}

, so no two inputs map to the same

b

. See Linear independence.

TF8. "

A x = b

solvable means

b

can be written as a linear combination of the rows of

A

False — it means

b

is a combination of the columns of

A

(the column view); rows live in a different space entirely.

TF9. "If

rank (A) = rank ([A ∣ b])

, the solution is automatically unique."

False — that only guarantees existence; uniqueness additionally needs that common rank to equal

n

, the number of unknowns.

Spot the error

SE1. "

A x = b \Rightarrow x = b / A

, by dividing both sides by

A

There is no matrix division; the legal move is left-multiplying by

A^{- 1}

(only when

A

is square and invertible), giving ==

x = A^{- 1} b

==.

SE2. "

A x = b

, so

x = b A^{- 1}

Order is wrong — you must left-multiply:

A^{- 1} A x = A^{- 1} b

gives

x = A^{- 1} b

. Matrix products don't commute, so

b A^{- 1}

isn't even well-shaped here.

SE3. "The system has 3 equations and 3 unknowns, so it has a unique solution."

Square shape alone guarantees nothing; you need

det A \neq = 0

(equivalently full rank). A singular

3 \times 3

system can have infinitely many or no solutions.

SE4. "

A

2 \times 3

, and

det A = 0

, so there's no unique solution."

The determinant is undefined for non-square matrices; use rank instead. Here

n = 3

but rank

\leq 2 < n

, so there can never be a unique solution regardless.

SE5. "Two solutions

x_{p}

and

x_{p}^{'}

exist, so the general solution is

x_{p} + x_{p}^{'}

Wrong structure — the general solution is

x_{p} + x_{h}

where

x_{h}

ranges over the null space; adding two particular solutions gives

A x = 2 b

, not

b

SE6. "rank

(A) = 2

, rank

([A ∣ b]) = 2

n = 4

, so there are

2

free parameters — namely rank many."

Free parameters are

n - r = 4 - 2 = 2

here by coincidence; the rule is ==

n - r

==, the number of columns minus rank, not the rank itself. (E.g.

r = 3, n = 4

gives

1

free parameter, not

3

SE7. "Since

A x = b

is consistent and

A x = 0

has only the trivial solution, we conclude nothing about uniqueness."

Actually this does force uniqueness: trivial null space + existence ⇒ exactly one solution, because any two solutions would differ by a null-space vector.

Why questions

WHY1. Why is matrix multiplication defined as "row-

i

dotted with

x

" rather than some other rule?

Because that exact definition makes

(A x)_{i} = \sum_{j} a_{ij} x_{j}

reproduce equation

i

verbatim, so

A x = b

is the original system compressed. See Matrix multiplication.

WHY2. Why does rank — and not the number of equations — decide existence and uniqueness?

Rank counts the number of independent constraints/directions; redundant rows add no information, so only independent constraints can pin down or contradict the unknowns.

WHY3. Why does

b

raising the augmented rank above

rank (A)

mean "no solution"?

It means

b

points in a direction the columns can't reach (a new independent direction), so no weighting

x

of the columns can ever build

b

—

b \in / Col (A)

WHY4. Why is the general solution

x_{p} + x_{h}

and not just

x_{p}

Because

A (x_{p} + x_{h}) = A x_{p} + A x_{h} = b + 0 = b

for any null-space vector

x_{h}

, so every particular solution is shifted by the whole null space. See Column space and null space.

WHY5. Why does

x = A^{- 1} b

only work for square invertible

A

A^{- 1}

exists only when

A

is square with independent columns; otherwise

A

either loses information (can't be undone) or isn't a one-to-one map, so no inverse exists.

WHY6. Why can we say

A x

is a "linear combination of columns" instead of computing dot products?

Grouping the terms

a_{ij} x_{j}

by column

j

gives

x_{j}

times column

a_{(j)}

, so

A x = \sum_{j} x_{j} a_{(j)}

— the weights are literally the unknowns.

WHY7. Why does having a nonzero null space destroy uniqueness but not existence?

A nonzero

x_{h}

can be added to any solution to make another, killing uniqueness; but it says nothing about whether

b

is reachable in the first place, which is a separate (existence) question.

Edge cases

EC1. What does

A x = b

describe when

A

is the zero matrix and

b = 0

Every vector

x

solves it (the constraint

0 = 0

is vacuous), so the solution set is all of

R^{n}

— the maximally-infinite case.

EC2. What if

A

is the zero matrix but

b \neq = 0

No solution —

A x = 0

for every

x

, so it can never equal a nonzero

b

; here

rank (A) = 0 < rank ([A ∣ b]) = 1

EC3. In the column view, what does

x = 0

(all weights zero) always produce?

The zero vector, since

\sum_{j} 0 \cdot a_{(j)} = 0

— which is why

A x = 0

is always consistent.

EC4. A single equation

x + y + z = 2

(so

m = 1, n = 3

): how many free parameters and why?

n - r = 3 - 1 = 2

free parameters — one plane in 3D is one constraint (rank 1), leaving two directions free to slide within the plane.

EC5. If

m = n

but two rows of

A

are identical, can the system be square and have infinitely many solutions?

Yes — identical rows drop the rank below

n

(so

det A = 0

), and if

b

is consistent with the dependence, you get infinitely many solutions despite the square shape.

EC6. What happens to the solution set of

A x = b

if you scale one whole equation by a nonzero constant

c

Nothing — multiplying a row of

[A ∣ b]

c \neq = 0

leaves both ranks and the solution set unchanged; it's a reversible row operation. See Gaussian elimination.

EC7. Can an

n \times n

homogeneous system

A x = 0

have exactly two solutions (and no more)?

No — solutions of

A x = 0

form a subspace, so it's either just

{0}

(one) or contains a nonzero vector and hence infinitely many (all its scalar multiples).

Connections

Systems of linear equations — matrix form Ax = b — the parent this bank drills.
Rank of a matrix — the single tool that resolves most traps here.
Column space and null space — existence lives in one, uniqueness in the other.
Determinant — square-only; the source of the " $det = 0 \Rightarrow$ no solution" trap.
Inverse of a matrix — legitimizes $x = A^{- 1} b$ and no other rearrangement.
Linear independence — independent columns $⟺$ trivial null space $⟺$ uniqueness.
Gaussian elimination — the row operations that never change the solution set.
Matrix multiplication — why $A x = b$ is the system, not a new rule.