Daily Math · 2026-06-18

Question 1

For general matrices $A$ and $B$ in $M_n(\mathbb{R})$ (with $n \ge 2$), which is true?

Accepted Answer

Matrix multiplication is **not** commutative in general. (Concrete example: $A = \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}$, $B = \begin{pmatrix} 0 & 0 \ 1 & 0 \end{pmatrix}$ — check $AB 
e BA$.)

Answer

$AB = BA$ always

Answer

$AB 
e BA$ in general

Answer

$AB$ does not exist

Answer

$A + B 
e B + A$

Question 2

The determinant $\det \begin{pmatrix} 1 & a & a^2 \ 1 & b & b^2 \ 1 & c & c^2 \end{pmatrix}$ equals (Vandermonde):

Accepted Answer

$\det = (b - a)(c - a)(c - b)$. (It's zero iff at least two of $a, b, c$ are equal.)

Answer

$abc$

Answer

$a + b + c$

Answer

$(b - a)(c - a)(c - b)$

Answer

$(a - b)(b - c)(c - a)$

Question 3

$(\mathbb{Z}, \cdot)$ — integers under multiplication — is:

Accepted Answer

Associative ✓, identity $1$ ✓. But $2$ has no integer multiplicative inverse. Hence a monoid but not a group.

Answer

a group

Answer

a monoid (but not a group)

Answer

a semigroup but not a monoid

Answer

not associative

Problems & worked solutions

Problem 1 — Matrices

Problem 2 — Determinants

Problem 3 — Semigroups & Monoids