Daily Math · 2026-06-27

Question 1

The trace of $A = \begin{pmatrix} 2 & 7 & 1 \ 0 & 3 & 4 \ 1 & 0 & 5 \end{pmatrix}$ is:

Accepted Answer

$\operatorname{tr}(A) = 2 + 3 + 5 = 10$.

Answer

$5$

Answer

$9$

Answer

$10$

Answer

$23$

Question 2

By the Rouché–Capelli theorem, a system has solutions iff:

Accepted Answer

A linear system is consistent (has at least one solution) iff $\operatorname{rank}(A) = \operatorname{rank}([A | b])$ — the rank of the coefficient matrix equals the rank of the augmented matrix.

Answer

$\det(A) = 0$

Answer

$\operatorname{rank}(A) = \operatorname{rank}([A | b])$

Answer

$\operatorname{rank}(A) > \operatorname{rank}([A | b])$

Answer

The system is square

Question 3

The function $f(x) = e^{-x}$ has at $x 	o \infty$:

Accepted Answer

$\lim_{x 	o \infty} e^{-x} = 0$, so $y = 0$ is a horizontal asymptote at $+\infty$.

Answer

a horizontal asymptote at $y = 0$

Answer

a horizontal asymptote at $y = 1$

Answer

a vertical asymptote at $x = 0$

Answer

no asymptote

Problems & worked solutions

Problem 1 — Matrices

Problem 2 — Systems of Linear Equations

Problem 3 — Asymptotes