Daily Math Problems — June 18, 2026: Matrices, Determinants, Algebra

Problems & worked solutions

RO M1

Problem 1 — Matrices

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

Show answer & worked solution

A. $A B = B A always$
B. $A B \neq B A in general$ ✓ correct
C. $A B does not exist$
D. $A + B \neq B + A$

Matrix multiplication is not commutative in general. (Concrete example: $A = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array})$ , $B = (\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array})$ — check $A B \neq B A$ .)

RO M1

Problem 2 — Determinants

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

Show answer & worked solution

A. $a b c$
B. $a + b + c$
C. $(b - a) (c - a) (c - b)$ ✓ correct
D. $(a - b) (b - c) (c - a)$

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

RO M1

Problem 3 — Semigroups & Monoids

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

Show answer & worked solution

A. $a group$
B. $a monoid (but not a group)$ ✓ correct
C. $a semigroup but not a monoid$
D. $not associative$

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

Problems & worked solutions

RO M1

Problem 1 — Matrices

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

Show answer & worked solution

A. $A B = B A always$
B. $A B \neq B A in general$ ✓ correct
C. $A B does not exist$
D. $A + B \neq B + A$

RO M1

Problem 2 — Determinants

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

Show answer & worked solution

A. $a b c$
B. $a + b + c$
C. $(b - a) (c - a) (c - b)$ ✓ correct
D. $(a - b) (b - c) (c - a)$

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

RO M1

Problem 3 — Semigroups & Monoids

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

Show answer & worked solution

A. $a group$
B. $a monoid (but not a group)$ ✓ correct
C. $a semigroup but not a monoid$
D. $not associative$

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