Types of matrices — row, column, square, diagonal, identity, zero, symmetric, skew-symmetric: Question bank

This is a reasoning gym for Types of Matrices. No heavy arithmetic here — every item hunts a misconception or a boundary case. Read the prompt, answer out loud, then reveal.

The vocabulary this page uses (built here, not borrowed)

Everything on this page runs on a handful of ideas, and we build each one right here so nothing depends on remembering another note.

Reading a cell's address. A matrix is a grid of numbers. To point at one cell we use a two-number address written as a subscript: $a_{ij}$ means "the number sitting in row $i$ , column $j$ ." The first subscript is the row count-down, the second is the column count-across. So $a_{23}$ is the entry in row $2$ , column $3$ . The whole matrix is often called $A$ , and its entries are the various $a_{ij}$ .

Order (shape). We describe a grid's shape as $rows \times columns$ . Throughout this page, when we need letters for a general shape we write $m$ for the number of rows and $n$ for the number of columns, so a matrix of order $m \times n$ has $m$ rows and $n$ columns. A third letter $p$ shows up only when we chain two matrices in a product.

The figure below is your visual anchor — the principal diagonal is the staircase of cells running top-left to bottom-right, drawn in blue.

Figure — Types of matrices — row, column, square, diagonal, identity, zero, symmetric, skew-symmetric

Definition The vocabulary, in plain words

Row matrix — a grid with exactly one row: order $1 \times n$ , e.g. $[2 - 57]$ . It is a single horizontal line of numbers.
Column matrix — a grid with exactly one column: order $m \times 1$ , e.g. $3 - 1 4$ . It is a single vertical stack of numbers.
Square matrix — a grid where the number of rows equals the number of columns: order $n \times n$ . Only these have a principal diagonal that reaches corner to corner.
Principal diagonal — the cells $a_{11}, a_{22}, a_{33}, \dots$ where the row number equals the column number (the blue staircase above). Everything else is off-diagonal (the orange cells).
Transpose $A^{T}$ — reflect the whole grid across that blue diagonal. Entry-wise, the cell in row $i$ , column $j$ swaps places with the cell in row $j$ , column $i$ , so $(A^{T})_{ij} = A_{j i}$ . See the mirror picture below.
$diag (d_{1}, \dots, d_{n})$ — shorthand for "build a square matrix, put $d_{1}, d_{2}, \dots, d_{n}$ down the blue diagonal, and fill every off-diagonal cell with $0$ ." So $diag (2, 5) = [2005]$ .
$O$ (zero matrix) — the matrix whose every entry is $0$ ; it comes in a shape for each order, e.g. $O_{2 \times 2} = [0000]$ .
$I$ (identity matrix) — the special square matrix with $1$ on every diagonal cell and $0$ everywhere off it: $I_{n} = diag (1, 1, \dots, 1)$ . For example $I_{2} = [1001]$ . It obeys $A I = I A = A$ .

The transpose is the single move behind symmetric and skew-symmetric, so look at exactly what it does — watch the $2$ and the $4$ trade places when we reflect across the diagonal:

Now apply that mirror move to a whole matrix and ask when the reflection lands somewhere special. The next figure puts the two special cases side by side:

Now the traps. Every item is beatable using only the boxed vocabulary above.

True or false — justify

TF1. Every diagonal matrix is symmetric.

True — off-diagonal entries are all

0

, so

a_{ij} = 0 = a_{j i}

automatically, meaning

A^{T} = A

TF2. Every symmetric matrix is diagonal.

False — symmetry only demands

a_{ij} = a_{j i}

; those mirrored pairs may be non-zero, e.g.

[1223]

is symmetric but not diagonal.

TF3. The identity matrix is a diagonal matrix.

True —

I = diag (1, \dots, 1)

has zeros off the diagonal and ones on it; "all ones" is just a special choice of diagonal entries.

TF4. A zero square matrix is both symmetric and skew-symmetric.

True —

O^{T} = O

(symmetric) and

O^{T} = - O

since

- O = O

(skew-symmetric); it is the only matrix that is both.

TF5. Every square matrix is either symmetric or skew-symmetric.

False — most square matrices are neither; e.g.

[1023]

has

A^{T} \neq = A

and

A^{T} \neq = - A

TF6. A skew-symmetric matrix must have all diagonal entries zero.

True — the condition

a_{ii} = - a_{ii}

forces

2 a_{ii} = 0

, so each diagonal entry is

0

TF7. Any matrix with a zero diagonal is skew-symmetric.

False — zero diagonal is necessary but not sufficient; you still need

a_{ij} = - a_{j i}

for the off-diagonal pairs, e.g.

[0520]

fails.

TF8. A

1 \times 1

matrix

[7]

is a row matrix, a column matrix, and a square matrix simultaneously.

True — it has one row, one column, and equal row/column count, so all three labels apply at once.

TF9. A row matrix can be symmetric.

False — symmetry needs

A^{T} = A

, but transposing a

1 \times n

row matrix (with

n > 1

) gives an

n \times 1

column matrix of a different shape, so equality is impossible unless

n = 1

TF10. If

A^{T} = A

and

A^{T} = - A

both hold, then

A = O

True — the two give

A = - A

, so

2 A = O

, hence

A = O

; the zero matrix is the only matrix that is simultaneously symmetric and skew-symmetric.

TF11. A diagonal matrix can equal the zero matrix.

True —

diag (0, 0, \dots, 0) = O

; "diagonal" allows any diagonal values, including all zeros.

Spot the error

SE1. "

[2 - 3 3 - 2]

is skew-symmetric because the off-diagonal entries are negatives of each other."

The off-diagonal check passes, but the diagonal entries

2

and

- 2

are non-zero, violating the mandatory

a_{ii} = 0

rule; it is not skew-symmetric.

SE2. "

diag (2, 0, - 5)

is not a diagonal matrix because one diagonal entry is

0

A zero on the diagonal is perfectly allowed; diagonal means only the off-diagonal entries must be zero, so this is a valid diagonal matrix.

SE3. "Since

[1005]

has zeros off the diagonal, it is the identity matrix."

It is diagonal, but the identity

I

additionally requires every diagonal entry to be

1

; a

5

on the diagonal disqualifies it.

SE4. "The matrix

[122435]

is symmetric because

a_{12} = a_{21} = 2

Symmetry is only defined for square matrices; this is

2 \times 3

(2 rows, 3 columns), so

A^{T}

3 \times 2

and can never equal

A

SE5. "

A O = O

so multiplying by the zero matrix always gives back the same-sized zero matrix as

A

The product

A O

is zero, but its shape follows the multiplication rule, so it need not match

A

's shape — for

A

of order

m \times n

(rows

\times

columns) times

O

of order

n \times p

, the result is

m \times p

SE6. "

A + O = A

works for any zero matrix

O

Only when

O

has the same order as

A

; addition requires matching dimensions, so

O

must be the zero matrix of

A

's exact size.

SE7. "A symmetric matrix stays symmetric under any relabelling, so its transpose is a different matrix."

For a symmetric matrix the transpose is literally the same matrix (

A^{T} = A

); transposing changes nothing.

Why questions

WQ1. Why does the diagonal of a skew-symmetric matrix carry no information?

Because every diagonal entry is forced to

0

a_{ii} = - a_{ii}

, so all the "content" lives strictly off the diagonal.

WQ2. Why is the identity matrix called the "1" of matrices?

Because

A I = I A = A

for any compatible

A

, mirroring how multiplying an ordinary number by

1

leaves it unchanged (the identity built above is exactly this "do-nothing" multiplier — see Matrix Operations).

WQ3. Why does symmetry require a square matrix?

The condition

A^{T} = A

compares

A

with its transpose; those two only have the same shape when the number of rows equals the number of columns.

WQ4. Why are diagonal matrices so easy to raise to powers?

Multiplying diagonal matrices just multiplies matching diagonal entries with no cross-terms, so

D^{k} = diag (d_{1}^{k}, \dots, d_{n}^{k})

— a key reason Diagonalization is prized.

WQ5. Why do only square matrices have a determinant and eigenvalues?

These notions describe how a matrix maps a space to itself (same input and output dimension), which requires equal rows and columns.

WQ6. Why is "

a_{ij} = a_{j i}

" enough to guarantee

A^{T} = A

Because

(A^{T})_{ij} = A_{j i}

by definition (the mirror reflection above), so if

A_{j i} = A_{ij}

for every pair, then

A^{T}

and

A

match entry-for-entry.

WQ7. Why is a diagonal matrix a "pure scaling" transformation geometrically?

Each coordinate axis is stretched by its diagonal entry with no mixing between axes, so nothing rotates or shears (see Linear Transformations).

WQ8. Why can a scaling inverse fail for a diagonal matrix?

If any diagonal entry is

0

, that axis is collapsed to nothing and cannot be undone, so the matrix has no inverse.

Edge cases

EC1. Is the empty-looking

2 \times 2

zero matrix

[0000]

diagonal?

Yes — all off-diagonal entries are zero, satisfying the diagonal condition; its diagonal entries simply happen to be zero too.

EC2. Can a matrix be a diagonal matrix and skew-symmetric at the same time?

Only the zero matrix — diagonal allows non-zero diagonal entries, but skew-symmetry forces them all to

0

, and the off-diagonal entries are already

0

EC3. What is the transpose of a

1 \times 1

matrix, and what does that make it?

1 \times 1

matrix equals its own transpose (nothing to flip), so it is automatically symmetric — and, being a single number, also trivially diagonal.

EC4. Is a single number treated as a matrix skew-symmetric?

1 \times 1

matrix is skew-symmetric only if its lone diagonal entry is

0

, i.e. only

[0]

qualifies.

EC5. Can a column matrix ever be symmetric?

Only if it is

1 \times 1

; for more than one row its transpose is a row matrix of a different shape, so

A^{T} \neq = A

EC6. If every entry of a square matrix equals the same constant

c

, when is it symmetric?

Always —

a_{ij} = c = a_{j i}

holds for all pairs, so a constant square matrix is symmetric for any

c

(though skew-symmetric only when

c = 0

EC7. Does a diagonal matrix with a repeated value like

diag (3, 3, 3)

have a special name?

Yes — it is a scalar matrix, equal to

3 I

; it scales every axis identically and behaves like the number

3

under multiplication.

Recall One-line self-test

Name a matrix that is symmetric, diagonal, and has an inverse. ::: Any $diag (d_{1}, \dots, d_{n})$ with all $d_{i} \neq = 0$ — e.g. the identity $I$ ; it is diagonal (hence symmetric) and invertible because no axis collapses.