Let A= 1 0 2 4; i^2 1 3 lg 100; 0 1 3 ln 1; sin(π)/(2) i^24 C_4^3 7. Find det(A).

bullet Evaluate each special entry: i^2 = -1, lg 100 = 2, ln 1 = 0 sin(π)/(2) = 1, i^24 = 1, C_4^3 = 4 bullet Assemble the numerical matrix: A = 1 0 2 4; -1 1 3 2; 0 1 3 0; 1 1 4 7 bullet Row-reduce. R_2 mathrel+= R_1, R_4 mathrel-= R_1: 1 0 2 4; 0 1 5 6; 0 1 3 0; 0 1 2 3 bullet R_3 mathrel-= R_2, R_4 mathrel-= R_2: 1 0 2 4; 0 1 5 6; 0 0 -2 -6; 0 0 -3 -3 bullet R_4 mathrel-= (3)/(2) R_3 gives upper-triangular with pivots 1, 1, -2, 6: det A = 1 · 1 · (-2) · 6 = -12

Daily Math Problems — May 19, 2026: Analytic Geometry, Matrices, Algebra

dailymath

Problems & worked solutions

RO M1

Problem 1 — Distances & Areas

The distance from the origin to the point $P (- 3, 4)$ is:

Show answer & worked solution

A. $1$
B. $5$ ✓ correct
C. $7$
D. $25$

$d = \sqrt{(- 3)^{2} + 4^{2}} = \sqrt{25} = 5$ .

RO M1

Problem 2 — Matrices

Let $A = (\begin{array}{cccc} 1 & 0 & 2 & 4 \\ i^{2} & 1 & 3 & \lg 100 \\ 0 & 1 & 3 & \ln 1 \\ \sin \frac{π}{2} & i^{24} & C_{4}^{3} & 7 \end{array})$ . Find $\det (A)$ .

Show answer & worked solution

A. $- 12$ ✓ correct
B. $1$
C. $- 3$
D. $6$
E. $- 1$
F. $3$

$∙$ Evaluate each special entry:

$i^{2} = - 1, \lg 100 = 2, \ln 1 = 0$

$\sin \frac{π}{2} = 1, i^{24} = 1, C_{4}^{3} = 4$

$∙$ Assemble the numerical matrix:

$A = (\begin{array}{cccc} 1 & 0 & 2 & 4 \\ - 1 & 1 & 3 & 2 \\ 0 & 1 & 3 & 0 \\ 1 & 1 & 4 & 7 \end{array})$

$∙$ Row-reduce. $R_{2} + = R_{1}$ , $R_{4} - = R_{1}$ :

$(\begin{array}{cccc} 1 & 0 & 2 & 4 \\ 0 & 1 & 5 & 6 \\ 0 & 1 & 3 & 0 \\ 0 & 1 & 2 & 3 \end{array})$

$∙$ $R_{3} - = R_{2}$ , $R_{4} - = R_{2}$ :

$(\begin{array}{cccc} 1 & 0 & 2 & 4 \\ 0 & 1 & 5 & 6 \\ 0 & 0 & - 2 & - 6 \\ 0 & 0 & - 3 & - 3 \end{array})$

$∙$ $R_{4} - = \frac{3}{2} R_{3}$ gives upper-triangular with pivots $1, 1, - 2, 6$ :

$\det A = 1 \cdot 1 \cdot (- 2) \cdot 6 = - 12$

RO M1

Problem 3 — Semigroups & Monoids

In a monoid $(M, *)$ with cancellation, if $a * b = a * c$ then:

Show answer & worked solution

A. $b = e (identity)$
B. $a is invertible$
C. $b = c$ ✓ correct
D. $b * c = e$

The cancellation law states: $a * b = a * c \Rightarrow b = c$ . (Even without inverses, this property may or may not hold; in a group it always does.)

Problems & worked solutions

RO M1

Problem 1 — Distances & Areas

The distance from the origin to the point $P (- 3, 4)$ is:

Show answer & worked solution

A. $1$
B. $5$ ✓ correct
C. $7$
D. $25$

$d = \sqrt{(- 3)^{2} + 4^{2}} = \sqrt{25} = 5$ .

RO M1

Problem 2 — Matrices

Let $A = (\begin{array}{cccc} 1 & 0 & 2 & 4 \\ i^{2} & 1 & 3 & \lg 100 \\ 0 & 1 & 3 & \ln 1 \\ \sin \frac{π}{2} & i^{24} & C_{4}^{3} & 7 \end{array})$ . Find $\det (A)$ .

Show answer & worked solution

A. $- 12$ ✓ correct
B. $1$
C. $- 3$
D. $6$
E. $- 1$
F. $3$

$∙$ Evaluate each special entry:

$i^{2} = - 1, \lg 100 = 2, \ln 1 = 0$

$\sin \frac{π}{2} = 1, i^{24} = 1, C_{4}^{3} = 4$

$∙$ Assemble the numerical matrix:

$A = (\begin{array}{cccc} 1 & 0 & 2 & 4 \\ - 1 & 1 & 3 & 2 \\ 0 & 1 & 3 & 0 \\ 1 & 1 & 4 & 7 \end{array})$

$∙$ Row-reduce. $R_{2} + = R_{1}$ , $R_{4} - = R_{1}$ :

$(\begin{array}{cccc} 1 & 0 & 2 & 4 \\ 0 & 1 & 5 & 6 \\ 0 & 1 & 3 & 0 \\ 0 & 1 & 2 & 3 \end{array})$

$∙$ $R_{3} - = R_{2}$ , $R_{4} - = R_{2}$ :

$(\begin{array}{cccc} 1 & 0 & 2 & 4 \\ 0 & 1 & 5 & 6 \\ 0 & 0 & - 2 & - 6 \\ 0 & 0 & - 3 & - 3 \end{array})$

$∙$ $R_{4} - = \frac{3}{2} R_{3}$ gives upper-triangular with pivots $1, 1, - 2, 6$ :

$\det A = 1 \cdot 1 \cdot (- 2) \cdot 6 = - 12$

RO M1

Problem 3 — Semigroups & Monoids

In a monoid $(M, *)$ with cancellation, if $a * b = a * c$ then:

Show answer & worked solution

A. $b = e (identity)$
B. $a is invertible$
C. $b = c$ ✓ correct
D. $b * c = e$

The cancellation law states: $a * b = a * c \Rightarrow b = c$ . (Even without inverses, this property may or may not hold; in a group it always does.)