Consider the following 2 x 2 matrix -5 A = 3. For any matrix, we have a clear understanding of some exponents: A" refers to multiplying a matrix to itself n times and A-l refers to the inverse matrix. Combining this idea gives A- = (A-')", the nth power of the inverse. A natural question to ask is, can we take the nth root of a matrix? We're going to try to find A/3, i.e. a matrix that when we cube it gives A. (a) Calculate the eigenvalues and eigenvectors of A. (b) Find a matrix P and a diagonal matrix D such that A = PDP-1. (c) Find a matrix E such that E = D. Explain why this result is simple compared to finding the cube root of A.

Consider the following 2 x 2 matrix -5 A = 3. For any matrix, we have a clear understanding of some exponents: A" refers to multiplying a matrix to itself n times and A-l refers to the inverse matrix. Combining this idea gives A- = (A-')", the nth power of the inverse. A natural question to ask is, can we take the nth root of a matrix? We're going to try to find A/3, i.e. a matrix that when we cube it gives A. (a) Calculate the eigenvalues and eigenvectors of A. (b) Find a matrix P and a diagonal matrix D such that A = PDP-1. (c) Find a matrix E such that E = D. Explain why this result is simple compared to finding the cube root of A.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Suppose the trace of a matrix is equal to 10. If this matrix is also an identity matrix, how many…

Q: (a) An n x n matrix B is a square root of a matrix A if B² = A. Show that the 2 x 2 identity matrix…

A: Given B is a square root of a matrix A

Q: Solve the matrix

A: we will solve such that we can get the solution to system of linear equations

Q: Solve the matrix equation AX = B for X. 3 1 34 *)-1] B= 42 5 6 X = A =

Q: ✓ 1.6 Each pair of matrices differ by one row operation. Use this operation to compare det(A) with…

Q: For the following matrix, do a single valid elementary row operation to get a 1 in row 1, column 1.…

Q: In the context of a 3 x 4 matrix, write down elementary matrices for the following operations: (a)…

Q: Solve the matrix equation Ax=b where A [ 2 -3 4 0 0 1 2 -3 b H 8 -10 16

Q: пр Select a square matrix (3 x 3 or larger) from anywhere in our textbook, outside of chapter 7.…

A: Let the given matrix be A=100010001 The given matrix is a 3×3 identity matrix To find (1)…

Q: MATRIX A: 4 -3 5 -11 9 8 -5 6 7 1 MATRIX B: -11 9 10 -5 3 MATRIX C: 0 4 -82 -48 -5 0 -2 0 0 0 -9 15…

Q: Which of the following is the solution for 剛 3] 长剧团-肉 = 0-2 - -5/2 3/2 №哟] -3/2] -5/2 3/2] 2-5/2…

Q: 3. Assume that A is an n × n matrix and A-¹ exists. Prove that (a) (A¹)−¹ – (A−¹)t. (b) (A²)−¹ –…

A: If A is an invertible matrix then AA-1=I (AB)T=BTAT At means interchange rows and columns in a…

Q: Given the matrices A and B, solve for X. 5X - B = A. 5 A: - [31]-[9] = and B = -4 Select one: a. O…

A: A = -35-3 -4 , B = -3 4-355X-B = A

Q: 7. Let A₁, A2, A3, A, be 3 x 2023 matrices and suppose A₂ R₁++R 2R₁+R₂, 3R₁ As A₁. (a) Write down a…

A: The elementary row Matrix E is obtained by applying the elementary row operation on the identity…

Q: Question -x1 + 3x₂ = 2 2x₁ + x₂ + 4x3 = 3 x2 + 2x3 = 4 [A-¹|I], Find A-¹. (a) Using [IA] (b) Find…

Q: For each of the matrices below, determine whether it is RREF. • If yes, copy the matrix on the…

A: A matrix is in RREF if it meets all of the following conditions: If there is a row where every…

Q: Solve for the matrix X: 35 1 2 *(²2)- ²×(²3)-( :) X = 72 20 The first row of X is given by is X11 =…

Q: Homework : Find the inverse of the matrix and verify that A-'A = A A-1 = 1,

Q: Suppose the trace of a matrix is equal to 10. If this matrix is also an identity matrix, how many…

A: Given: The trace of a matrix is equal to 10 and the matrix is an identity matrix. To determine:…

Q: can you put the answer in matrix form

Q: Write equations z + 2y + 3: = a, y – z - b, and 2y + 2: - c, as the matrix equation ()-) A and then:…

Q: I just need to know how to solve part b of the problem with the given equation. I tried, but I could…

Q: Assuming that a matrix, A, has an inverse, which of the following is NOT true A. A can be…

Q: Use the of a the inverse Possible. w/or O matrix capabilities calculator to find the matrix, of 2…

A: Find the inverse of the given matrix.

Q: Exercise 5: Determine if the following series converges or diverges. If it converges, find its sum.…

Q: Solve the matrix equation for x -2 1 4 3 2 1 -1 -1| X = 6. 7 8. 9. 1 1 -4 1 3 7 9.

A: The solution is given in the next steps.

Q: To divide the LHS by a matrix on the RHS, what steps do you follow?

A: To find: To divide the LHS by a matrix on the RHS, what steps do you follow?

Q: The equation A² - 2A + I = O implies that ---Select--- Need Help? Read It ✓. It follows that…

Q: Solve the following: 3 1 a. Find the inverse matrix (A") if the matrix A = -1 -4 3

Q: (ЗАТ + 4CТВ)Т- 54 - 4B

Q: Here matrix A 4 2- i 9 A= |2 +i 0 3+i 3- i 7 Can you solve for A· A" (Transpose matrix)?

A: Explanation of the answer is as follows

Question

100%

Consider the following 2 x 2 matrix
A =
For any matrix, we have a clear understanding of some exponents: A" refers to multiplying a matrix to
itself n times and A-l refers to the inverse matrix. Combining this idea gives A-n = (A-!)", the nth
power of the inverse.
A natural question to ask is, can we take the nth root of a matrix? We're going to try to find A/3, i.e. a
matrix that when we cube it gives A.
(a) Calculate the eigenvalues and eigenvectors of A.
(b) Find a matrix P and a diagonal matrix D such that A = PDP-1.
(c) Find a matrix E such that E = D. Explain why this result is simple compared to finding the cube
root of A.
(d) Using part (c), find a matrix B such that B3 = A.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step 8

Trending now

This is a popular solution!

Step by step

Solved in 8 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Inverse of a Matrix$

Learn more about

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

Similar questions

*Consider the following matrices 1 3 A=(-₂3) B=(²5) C- (₁1/²82) D= 3 0 2, (d) Decide which of the matrices can be squared and calculate the squares. (e) Calculate (if possible) A-1 and B-¹, (f) What is E-¹? Attempt only only if you can solve all parts. I'll upvote your answer 0 0 0 c = () -- 6:9) E 0 1 0 001
I need the answer quickly
Explain how to solve the matrix equation AX = B.
What is the procedure for obtaining the inverse of a matrix?
Please show solution
The matrix A = (13¹) is closely related to the imaginary number i. (a) Calculate A², A³, A4, expressing each as a multiple of either A or the identity matrix I. Infer the general pattern for A4, A4n+1, A4n+2 A4n+3, where n is an integer, and notice how this compares to powers of i (b) The exponential of a matrix is defined in terms of the power series eX = 1 + X + ² X ²X² + ²/1 ² Calculate e, evaluating the series to obtain a simple 2 x 2 matrix. Explain the result geometrically, noting the close analogy with e ·X³+.
How do you calculate when the matrix is equal to 0 ?