Properties — row operations, multiplicativity: Question bank

Quick symbol reminder so nothing below is used unexplained:

$det A$ = the signed volume-scaling factor of the map $A$ (how much the unit cube's volume grows, negative if orientation flips). See Volume, Orientation & the Determinant.
A matrix is $n \times n$ (square, $n$ rows and $n$ columns).
Elementary matrix $E$ = the identity with one row operation applied; multiplying by it is that operation. See Elementary Matrices.
Singular = not invertible = $det = 0$ . See Invertibility & Singular Matrices.

True or false — justify

TF1. Swapping two rows twice returns the determinant to its original value.

True. Each swap multiplies

det

- 1

, and

(- 1) (- 1) = 1

, so two swaps undo each other.

TF2. Scaling a row by

0

is a legal elementary operation.

False. The scale operation requires

k \neq = 0

; scaling by

0

makes a zero row (and cannot be undone), forcing

det = 0

— it destroys information rather than transforming it.

TF3.

det (A + B) = det A + det B

for all square

A, B

False. Multilinearity is linear in one row at a time, not in the whole matrix. Counterexample:

A = B = I_{2}

gives

det (2 I_{2}) = 4 \neq = 1 + 1

TF4.

det (k A) = k det A

for an

n \times n

matrix.

False. Scaling

A

k

scales all $n$ rows, so

det (k A) = k^{n} det A

; only for

n = 1

do these agree.

TF5. If

det A = 0

then

det (A B) = 0

for every

B

of matching size.

True. By multiplicativity

det (A B) = det A det B = 0 \cdot det B = 0

; a singular map collapses volume no matter what follows.

TF6. Row operations change the determinant, so column operations must too.

True, but they follow the same rules — because

det (A^{T}) = det A

, and transposing turns rows into columns. A column swap flips sign, a column scale multiplies by

k

, etc.

TF7.

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

False. From

det A \cdot det (A^{- 1}) = det I = 1

we get

det (A^{- 1}) = 1/ det A

, the reciprocal, not the negative.

TF8. A replacement

R_{i} \to R_{i} + k R_{j}

can turn a singular matrix into an invertible one.

False. Replacement leaves

det

unchanged, so if

det

was

0

it stays

0

— singularity is invariant under replacement (see Gaussian Elimination & Row Echelon Form).

TF9. Every matrix can be written as a product of elementary matrices.

False. Only invertible matrices factor this way; a singular matrix cannot, since a product of elementaries always has nonzero determinant.

TF10. If

A

has a row of all zeros, then

det A = 0

True. Scaling that zero row by any

k

leaves it unchanged, yet must multiply

det

k

; the only number satisfying

det = k det

for all

k

0

Why questions

WHY1. Why does swapping two rows flip the sign, without adding a new axiom?

Feeding equal rows

u + v

into both slots gives

0

; expanding by linearity leaves

D (u, v) + D (v, u) = 0

, so

D (v, u) = - D (u, v)

— the sign flip is forced by "equal rows give 0" plus multilinearity. See Multilinear Alternating Forms.

WHY2. Why is replacement the "safe" operation for Gaussian elimination?

Because it leaves

det

exactly unchanged, you can clear out entries to reach echelon form without ever having to track a correction factor.

WHY3. Why is the determinant of a triangular matrix just the product of the diagonal?

Cofactor expansion down the first column leaves only one surviving nonzero term at each stage, recursively multiplying the diagonal entries together. See Determinant — Definition (cofactor / Leibniz).

WHY4. Why does multiplicativity

det (A B) = det A det B

hold geometrically?

B

scales volume by

det B

, then

A

scales that result by

det A

; composing the two maps multiplies the scaling factors, and sign flips (orientation) multiply the same way.

WHY5. Why does the algebraic proof of multiplicativity split into invertible and singular cases?

Only invertible

A

factor as a product of elementaries

E_{1} \dots E_{m}

; the singular case needs a separate argument, using that

det A = 0

and

A B

inherits the rank deficiency so

det (A B) = 0

too.

WHY6. Why does

det (A^{T}) = det A

let us apply every row rule to columns?

A column operation on

A

is a row operation on

A^{T}

; since transposing does not change the determinant, the effect on

det

is identical.

WHY7. Why must

det (A^{- 1}) = 1/ det A

and why does this prove invertible matrices have nonzero determinant?

From

det A \cdot det (A^{- 1}) = det I = 1

, the product is

1

, so neither factor can be

0

— hence

det A \neq = 0

is necessary for an inverse to exist.

Edge cases

EC1. What is

det

of the

n \times n

zero matrix, and why?

0

== — every row is a zero row, and a zero row forces

det = 0

(scaling it by

k

can't change it yet must multiply

det

k

EC2. If two rows of

A

are identical, what is

det A

0

== directly by the alternating property — the defining rule of the determinant is that equal rows give

0

EC3. If one row is a scalar multiple of another, is

det A

zero?

Yes. A replacement

R_{i} \to R_{i} - c R_{j}

(unchanging

det

) can make row

i

all zeros, and a zero row gives

det = 0

EC4. For a

1 \times 1

matrix

[a]

, does

det (k A) = k^{n} det A

still hold?

Yes, trivially — with

n = 1

it reads

k a = k^{1} \cdot a

, the one case where

det (k A) = k det A

is correct.

EC5. What is

det

of a permutation matrix that swaps rows an odd number of times?

- 1

== — each of the odd swaps multiplies the identity's determinant

1

- 1

, netting

(- 1)^{odd} = - 1

EC6. If

A

is a real matrix with

det A < 0

, what does the sign mean geometrically?

The map

A

reverses orientation (it flips the shape like a mirror) while scaling volume by

∣ det A ∣

. See Volume, Orientation & the Determinant.

EC7. If

det A = 1

and

λ

ranges over the eigenvalues, what constraint does that place on them?

Their product is $1$ , since

det A = \prod λ_{i}

; this is why volume-preserving maps have eigenvalues multiplying to

1

. See Eigenvalues — det as product of eigenvalues.

EC8. Can a replacement operation ever change the determinant if you accidentally use

i = j

(adding a multiple of a row to itself)?

The rule requires

i \neq = j

; adding

k R_{i}

R_{i}

itself is really a scaling

R_{i} \to (1 + k) R_{i}

, which multiplies

det

(1 + k)

— so watch the index.

Question bank — Properties — row operations, multiplicativity

True or false — justify

Spot the error

Why questions

Edge cases