University:
Carleton College
Course:
MATH 120 | Calculus 2
Academic year:

2024
Views:

299

Pages:

3
Author:

Nevaeh Bauer

Question 1 Row equivalence. What is it exactly? When matrices are row equivalent... why is this important? If a matrix like: [1 0] [-3 1] is row equivalent to the identity matrix (add 3 times the first row to the second), what does that mean exactly? Why is this a concept that we have to know as students of linear algebra? These matrices aren't Answer 1 (1) [1 0] [-3 1] R₂ + 3R₁ [1 0] [0 1] (2) The information is conveyed via the related system of homogeneous equations above. x = 0 -3x + y = 0 → y=0 ⇒ A matrix's row space is unaffected by simple row operations. In particular, the row space is the same for any two row-equivalent matrices. ⇒ If and only if they are equivalent, two matrices of reduced row echelon form will share the same row space. Answer 2 The solutions to the two matrix equations Ax=b and Bx=b are the same as long as A and B are row equivalent, which is why I believe the idea of row equivalence to be crucial. It is frequently desirable to convert a matrix problem Ax=b into an equation Bx=b that is simpler to solve since row operations do not alter the solution set. Question 2 True or False If A and B are n x n lower triangular matrices, then AB is also lower triangular Answer Consider a 2 x 2 lower triangular matrix [ a 0 ] A = [ b c ] and B = [ d 0 ] [ e f ] Now taking the multiplication of A and B, we get, [ a 0 ] [ d 0 ] = [ ad 0 ] AB = [ b c ] [ e f ] [ bd + ce cf ] Therefore, AB is also an upper triangular matrix, which is N Hence, if A and B are n x n lower triangular matrices, then AB is also lower triangular. Therefore, the given statement is true. Question 3 Penguins spend almost 75% of their lives in the sea. An Emperor Penguin in the wild has a life span of about 18 years. About how many years does this penguin spend in the sea? Answer Since penguins spend almost 75% of their lives in the sea, then we multiply the life of an Emperor Penguin by 0.75 (the decimal form of 75%) to find how many years it spends in the sea: 0.75 × 18 = 13.5 years Question 4 A scatterplot shows a moderately strong negative linear relationship. Use your calculator or software to find the equation of the least-squares regression line. Answer y = 479.93 - 9.69x Question 5 Determine whether each of the following statements is true or false, and explain why. If A and B are square matrices of the same size, then AB = BA Answer Given, If A and B are identical-size square matrices, then AB = BA. Step 2 The assertion made is untrue. For example, Let A = [1 0] and B = [2 4] [2 3] [1 3] Then, AB = [1 0] [2 4] = [ 2 4] [2 3] [1 3] [ 7 17] and BA = [2 4] [1 0] = [ 10 12] [1 3] [2 3] [ 7 9] Clearly, AB ≠ BA