Explanation Theorem 1: Consider A is a n × n matrix. The Determinants of the matrix A can be computed using cofactor expansion across the rows or down the columns. Show the expression for computing the Determinant of matrix A using cofactor expansion across the i t h row as follows: det A = a i 1 C i 1 + a i 2 C i 2 + ⋯ + a i n C i n (1) Here, C i j is the cofactor of matrix A . Show the expression for the cofactor of matrix A as follows C i j = ( − 1 ) i + j det A i j (2) Here, i and j are the row and column position of elements, A i j is the sub matrix obtained by deleting i t h row and j t h column of matrix A . Modify Equation (1) using Equation (3). det A = a i 1 ( − 1 ) i + 1 det A i 1 + a i 2 ( − 1 ) i + 2 det A i 2 + ⋯ + a i n ( − 1 ) i + n det A i n (3) Given information: The line passes through the point ( x 1 , y 1 ) with slope m . Refer Exercise 7. The Equation of the line passing through the point ( x 1 , y 1 ) and ( x 2 , y 2 ) is shown as follows: det [ 1 x y 1 x 1 y 1 1 x 2 y 2 ] = 0 (4) Calculation: Consider the line passes through the point ( x 1 , y 1 ) with slope m . Consider a point ( x 2 , y 2 ) also lies in the line such that ( x 2 , y 2 ) = ( x 1 + 1 , y 1 + m ) Modify Equation (4) using the points ( x 1 , y 1 ) and ( x 1 + 1 , y 1 + m ) as follows: det [ 1 x y 1 x 1 y 1 1 x 1 + 1 y 1 + m ] = 0 Consider the row 1, row 2 and row 3 of the matrix is denoted by R1 , R2 and R3. Replace R2 by R3 − R2 on left hand side of the Equation. det [ 1 x y 1 x 1 y 1 0 1 m ] R3 → R3 − R2 = 0 (5) Calculate the left hand side of Equation (5) by cofactor expansion across the first row as follows: Substitute 1 for i and 3 for n in Equation (3)

Linear Algebra and Its Applications (5th Edition)

5th Edition

ISBN: 9780321982384

Author: David C. Lay, Steven R. Lay, Judi J. McDonald

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 3, Problem 8SE

To determine

To find: a determinant Equation of the line passing through the points (x1,y1) and with slope m.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 3, Problem 7SE

Chapter 3, Problem 9SE

Students have asked these similar questions

On Feb. 8, this year, at 6am in the morning all UiB meteorology professors met to discuss a highly unfortunate and top-urgent crisis: Their most precious instrument, responsible for measuring the air temperature hour-by- hour, had failed - what if the Bergen public would find out? How would they plan their weekend without up-to-date air temperature readings? Silent devastation - and maybe a hint of panic, also - hung in the room. Apprentice Taylor, who - as always - was late to the meeting, sensed that this was his chance to shine! Could they fake the data? At least for some hours (until the measurements would work again)? He used to spend a lot of time online and thus knew the value of fake data, especially when it spread fast! He reminded the crying professors of a prehistoric project with the title "Love your derivatives as you love yourself!" - back then, they had installed top-modern technology that not only measured the air temperature itself, but also its 1st, 2nd, 3rd, 4th, and…

Consider a forest where the population of a particular plant species grows exponentially. In a real-world scenario, we often deal with systems where the analytical function describing the phenomenon is not available. In such cases, numerical methods come in handy. For the sake of this task, however, you are provided with an analytical function so that you can compare the results of the numerical methods to some ground truth. The population P(t) of the plants at time t (in years) is given by the equation: P(t) = 200 0.03 t You are tasked with estimating the rate of change of the plant population at t = 5 years using numerical differentiation methods. First, compute the value of P'(t) at t = 5 analytically. Then, estimate P'(t) at t = 5 years using the following numerical differentiation methods: ⚫ forward difference method (2nd-order accurate) 3 ⚫ backward difference method (2nd-order accurate) ⚫ central difference method (2nd-order accurate) Use h = 0.5 as the step size and round all…

Nicole organized a new corporation. The corporation began business on April 1 of year 1. She made the following expenditures associated with getting the corporation started: Expense Date Amount Attorney fees for articles of incorporation February 10 $ 40,500 March 1-March 30 wages March 30 6,550 March 1-March 30 rent Stock issuance costs March 30 2,850 April 1-May 30 wages Note: Leave no answer blank. Enter zero if applicable. April 1 May 30 24,000 16,375 c. What amount can the corporation deduct as amortization expense for the organizational expenditures and for the start-up costs for year 1 [not including the amount determined in part (b)]? Note: Round intermediate calculations to 2 decimal places and final answer to the nearest whole dollar amount. Start-up costs amortized Organizational expenditures amortized

Chapter 3 Solutions

Linear Algebra and Its Applications (5th Edition)

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

a determinant Equation of the line passing through the points ( x 1 , y 1 ) and with slope m .

Solution Summary: The author explains the determinant Equation of the line passing through the points left,y_1right, and with slope m. The Determinants of matrix A can be computed using

Videos

To find: a determinant Equation of the line passing through the points (x1,y1) and with slope m.

Want to see the full answer?

Chapter 3 Solutions