Math Problems 7/31/26

Question 1

Let \(A\) be a \(3 \times 3\) matrix with \(det(A) = 10\). A new matrix B is formed by performing the following sequence of row operations on \(A\).

1. \(R_1 \to R_3\)

2. \(R_2 \to 4 R_2\)

3. \(R_1 \to R_1 - 2 R_2\)

What is \(det(B)\)?

\[A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 10 \end{bmatrix}\]

# Define our example matrix A
A <- matrix(data = c(1,0,0,0,1,0,0,0,10),nrow = 3,ncol = 3,byrow = T)

# A. Swap Row 1 and Row 3
B <- A
B[c(1,3), ] <- B[c(3,1), ]

# B. R2 -> 4 R2
B[2, ] <- 4 * B[2, ]

# C. R1 -> R1 - 2 R2
B[1, ] <- B[1, ] - 2 * B[2, ]

# Display matrix B
cat("Matrix B: \n")

## Matrix B:

print(B)

##      [,1] [,2] [,3]
## [1,]    0   -8   10
## [2,]    0    4    0
## [3,]    1    0    0

# Determinant of matrix B
cat("det(B) =",det(B),"\n")

## det(B) = -40

Question 2

You are given the points A(2,3), B(4,7), C(6,5). Plot these points to make a polygon.

# install.packages("tidyverse")
library(tidyverse)

## Warning: package 'lubridate' was built under R version 4.5.2

## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr     1.1.4     ✔ readr     2.1.5
## ✔ forcats   1.0.1     ✔ stringr   1.5.2
## ✔ ggplot2   4.0.0     ✔ tibble    3.3.0
## ✔ lubridate 1.9.4     ✔ tidyr     1.3.1
## ✔ purrr     1.1.0     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

q2_data <- data.frame(Point = c("A","B","C"),
                      X = c(2,4,6),
                      Y = c(3,7,5))
ggplot(q2_data,aes(x = X,y = Y)) +
  geom_polygon(fill = "lightblue",alpha = 0.5,color = "black",lwd = 1.25) +
  geom_point(size = 3) +
  geom_text(aes(label = Point),nudge_y = 0.3,fontface = "bold") +
  theme_gray()

Question 3

Ursula throws two tetrahedral dice at the same time and notes the scores on their bottom faces. Answer the following parts.

A. What is the probability both numbers are prime?

# install.packages("pracma")
library(pracma)

## 
## Attaching package: 'pracma'

## The following object is masked from 'package:purrr':
## 
##     cross

die1 <- 1:4 # four sides to a tetrahedral die
die2 <- 1:4 # four sides to a tetrahedral die
N <- 1e6 # 1 million trials
counter1 <- 0
for (i in 1:N) {
  roll1 <- sample(x = die1,size = 1,replace = T)
  roll2 <- sample(x = die2,size = 1,replace = T)
  if (isprime(roll1) & isprime(roll2)) {
    counter1 <- counter1 + 1
  }
}
probability1 <- counter1 / N
cat("The probability both numbers are prime is:",probability1,"\n")

## The probability both numbers are prime is: 0.249884

B. What is the probability both numbers are not prime?

# install.packages("pracma")
library(pracma)
die3 <- 1:4
die4 <- 1:4
N2 <- 1e6
counter2 <- 0
for (j in 1:N2) {
  roll3 <- sample(x = die3,size = 1,replace = T)
  roll4 <- sample(x = die4,size = 1,replace = T)
  if (!isprime(roll3) & !isprime(roll4)) {
    counter2 <- counter2 + 1
  }
}
probability2 <- counter2 / N2
cat("The probability both numbers are not prime is:",probability2,"\n")

## The probability both numbers are not prime is: 0.249687