Matrices — review, operations, types: Question bank

Reminder of the vocabulary these questions lean on: a matrix is a grid of numbers standing for a linear transformation; the product $A B$ means "do $B$ first, then $A$ "; the transpose $A^{⊤}$ swaps rows and columns; singular means $det A = 0$ (no inverse); orthogonal means $A^{⊤} A = I$ . Every answer below re-explains the "why", so you never need to have memorised these.

True or false — justify

Matrix addition is commutative:

A + B = B + A

for matrices of the same order.

True — addition is entrywise, and each number-pair

a_{ij} + b_{ij}

obeys ordinary number commutativity. The "do B first" ordering issue only appears in multiplication.

Matrix multiplication is commutative.

False —

A B

means "do

B

then

A

"; swapping to

B A

changes the order of actions (like socks-then-shoes vs shoes-then-socks), so generally

A B \neq = B A

A B

exists, then

B A

must also exist.

False —

A B

needs columns of

A

= rows of

B

. A

2 \times 3

times a

3 \times 2

gives

2 \times 2

, but the reverse is

3 \times 2

times

2 \times 3 = 3 \times 3

: existing but a different shape, so they can't even be equal.

Every square matrix commutes with the identity:

A I = I A = A

True —

I

is the "do nothing" transformation, so applying it before or after

A

changes nothing. Identity is the one matrix that always commutes with everything.

A^{⊤} = A

and

B^{⊤} = B

, then

A B

is symmetric.

False —

(A B)^{⊤} = B^{⊤} A^{⊤} = B A

, which equals

A B

only if

A

and

B

commute. Two symmetric matrices need not commute, so their product is usually not symmetric.

The diagonal of any skew-symmetric matrix is all zeros.

True — skew means

a_{ij} = - a_{j i}

, so on the diagonal

a_{ii} = - a_{ii}

, forcing

2 a_{ii} = 0

, hence

a_{ii} = 0

Every orthogonal matrix is invertible.

True —

A^{⊤} A = I

means

A^{⊤}

is the inverse. Orthogonal matrices are rotations/reflections; they preserve lengths and can always be undone.

det A = 0

then

A

has no inverse.

True — the determinant measures how the transformation scales area/volume; a zero means it collapses space onto a lower dimension, and you cannot un-flatten it. See Determinant and Inverse of a Matrix.

λ (A B) = (λ A) B = A (λ B)

for a scalar

λ

True — a scalar scales the whole transformation's output uniformly, so it can slide freely to either factor. Scalars carry no "order", unlike matrices.

The transpose of a sum is the sum of transposes:

(A + B)^{⊤} = A^{⊤} + B^{⊤}

True — transposing just relabels

(i, j) \to (j, i)

entry by entry, and addition is entrywise, so the two operations don't interfere.

Spot the error

(A B)^{⊤} = A^{⊤} B^{⊤}

, since transpose distributes over products."

Wrong — the correct reversal law is

(A B)^{⊤} = B^{⊤} A^{⊤}

. Transpose reverses roles and order (socks-and-shoes);

A^{⊤} B^{⊤}

usually has mismatched inner dimensions and doesn't even exist.

A B = 0

, so either

A = 0

B = 0

Wrong — matrices have zero divisors. E.g.

[1000] [0001] = 0

with neither factor zero. Cancellation only holds for invertible matrices.

A B = A C

implies

B = C

Wrong — you may only cancel

A

when

A

is invertible. If

A

is singular it can collapse different inputs to the same output, so

B \neq = C

is possible.

"Multiplying matrices means multiplying corresponding entries, just like adding."

Wrong — that's the Hadamard product, which doesn't compose transformations. Real multiplication is (row of

A

)·(column of

B

) as a dot product, forced by

(A B) x = A (B x)

(A + B)^{2} = A^{2} + 2 A B + B^{2}

for matrices."

Wrong — expanding gives

A^{2} + A B + B A + B^{2}

, and

A B \neq = B A

in general, so the middle can't be combined into

2 A B

. The binomial shortcut needs commutativity.

"Since

A^{2} = I

, matrix

A

must be the identity."

Wrong — reflections also square to

I

(reflect twice = original). E.g.

[10 0 - 1]

squares to

I

but isn't

I

. See Eigenvalues and Eigenvectors (eigenvalues

\pm 1

2 \times 3

matrix can be added to a

3 \times 2

matrix by lining them up."

Wrong — addition pairs entry

a_{ij}

with

b_{ij}

, which only exists when both dimensions match. Different shapes have no natural pairing.

A^{⊤} A

and

A A^{⊤}

are the same matrix."

Wrong — for a non-square

A

they even have different sizes (

n \times n

m \times m

), and for square

A

they generally differ unless

A

is normal. Order matters here too.

Why questions

Why must the inner dimensions match for

A B

to exist?

Because entry

(A B)_{ij}

is a dot product of a row of

A

with a column of

B

; a dot product needs the same number of terms on both sides, i.e. columns of

A

= rows of

B

Why is matrix multiplication defined by row·column and not entrywise?

So that

(A B) x = A (B x)

— the product must represent doing transformation

B

then

A

. Entrywise multiplication would not compose maps or solve linear systems.

Why does the transpose reverse the order of a product?

Transpose swaps rows and columns, effectively viewing the map "from the output side". Composition read backwards flips both the roles and the order, giving

(A B)^{⊤} = B^{⊤} A^{⊤}

Why does every square matrix split uniquely into a symmetric plus a skew-symmetric part?

Write

A = \frac{1}{2} (A + A^{⊤}) + \frac{1}{2} (A - A^{⊤})

: the first piece equals its own transpose (symmetric), the second equals minus its transpose (skew), and the split is forced.

Why does

det A = 0

signal no inverse?

The determinant is the area/volume scaling factor of the transformation. Zero means space is squashed flat, so distinct inputs merge — that collapse can't be reversed, hence no inverse.

Why do orthogonal matrices preserve lengths?

A^{⊤} A = I

, then

∣ A x ∣^{2} = (A x)^{⊤} (A x) = x^{⊤} A^{⊤} A x = x^{⊤} x = ∣ x ∣^{2}

. The length is unchanged, so it's a rotation or reflection.

Why is the identity matrix the multiplicative "1" of matrices?

It's the "do nothing" transformation:

I A = A I = A

. Just as multiplying a number by 1 leaves it unchanged, composing with

I

leaves any map unchanged.

Why can

A B

be non-commutative even when both are diagonal?

It isn't — two diagonal matrices do commute, since each just scales independent axes and scaling factors multiply order-freely. Non-commutativity needs mixing of axes (rotation, shear).

Edge cases

What is the product of a

1 \times n

row matrix with an

n \times 1

column matrix?

1 \times 1

matrix — a single number, exactly the dot product of the two vectors. This is why matrix product entries are dot products.

Is the

1 \times 1

matrix

[c]

symmetric, skew-symmetric, or both?

Symmetric always (

[c]^{⊤} = [c]

); skew-symmetric only if

c = 0

, since skew forces the (only) diagonal entry to vanish. The zero

1 \times 1

matrix is both.

Can a matrix be both symmetric and skew-symmetric?

Yes, but only the zero matrix.

A = A^{⊤}

and

A = - A^{⊤}

give

A = - A

, so

2 A = 0

, hence

A = 0

Is the zero matrix orthogonal?

No — orthogonality requires

A^{⊤} A = I

, but for the zero matrix

A^{⊤} A = 0 \neq = I

. The zero map collapses everything and preserves no lengths.

What happens when you multiply any matrix by the zero matrix (of compatible size)?

You get the zero matrix — every output entry is a dot product with an all-zero row or column. This holds even though neither factor being zero was required for a zero product elsewhere.

Is a

1 \times 1

matrix's multiplication commutative?

Yes — it's just ordinary number multiplication,

[a] [b] = [ab] = [b] [a]

. Non-commutativity is a genuinely higher-dimensional phenomenon.

Does every non-zero matrix have an inverse?

No — only square, non-singular matrices do. Non-square matrices and square matrices with

det = 0

(e.g. any matrix with a zero row) have no two-sided inverse.

A

3 \times 3

with two identical rows, what is

det A

, and is

A

invertible?

det A = 0

, so

A

is singular and non-invertible — identical rows mean the transformation collapses a dimension, squashing volume to zero.

Recall One-line summary of the traps

Entrywise for add/scale, dot-products for multiply; order matters ( $A B \neq = B A$ , $(A B)^{⊤} = B^{⊤} A^{⊤}$ ); cancel only when invertible; $det = 0$ collapses space; zero products don't force zero factors.

Connections

Matrices — review, operations, types — the parent topic these traps stress-test.
Determinant and Inverse of a Matrix — singularity and cancellation.
Linear Transformations — why non-commutativity is about ordered actions.
Systems of Linear Equations — why entrywise multiply would fail.
Eigenvalues and Eigenvectors — reflections squaring to $I$ .
Dot Product and Vectors — each product entry is a dot product.