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1. (a) Explain what it means for the vectors $v_1, \dots, v_m$ in $\mathbb{R}^n$ to be linearly independent. (b) Let $A = \begin{pmatrix} 1 & 7\\ 2 & 14 \end{pmatrix}$. (i) Find the characteristic polynomial of A, and the eigenvalues of A. (ii) Find 5 x 5 matrices $C_1$ and $C_2$ that have the same nonzero eigenvalues as A. In addition, $C_1$ should have all its diagonal elements equal to 3, and $C_2$ should have the diagonal elements equal to 1, 2, 3, 4, 5.

          1. (a) Explain what it means for the vectors $v_1, \dots, v_m$ in $\mathbb{R}^n$ to be linearly independent.
(b) Let
$A = \begin{pmatrix} 1 & 7\\ 2 & 14 \end{pmatrix}$.
(i) Find the characteristic polynomial of A, and the eigenvalues of A.
(ii) Find 5 x 5 matrices $C_1$ and $C_2$ that have the same nonzero eigenvalues as
A. In addition, $C_1$ should have all its diagonal elements equal to 3, and
$C_2$ should have the diagonal elements equal to 1, 2, 3, 4, 5.

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1. (a) Explain what it means for the vectors v1, …, vm in ℝ^n to be linearly independent.
(b) Let
A =
< p m a t r i x >.
(i) Find the characteristic polynomial of A, and the eigenvalues of A.
(ii) Find 5 x 5 matrices C1 and C2 that have the same nonzero eigenvalues as
A. In addition, C1 should have all its diagonal elements equal to 3, and
C2 should have the diagonal elements equal to 1, 2, 3, 4, 5.

Added by Scott S.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Adi S

05/09/2024

Step 1

Step 1: The characteristic polynomial of a matrix A is given by det(A - λI), where λ is a scalar and I is the identity matrix. Show more…

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1. (a) Explain what it means for the vectors V1...., Vm in R" to be linearly inde- pendent. (b) Let $$A = \begin{pmatrix} 1 & 7 \\ 2 & 14 \end{pmatrix}.$$ (i) Find the characteristic polynomial of A, and the eigenvalues of A. (ii) Find 5 x 5 matrices C₁ and C₂ that have the same nonzero eigenvalues as A. In addition, C₁ should have all its diagonal elements equal to 3, and C₂ should have the diagonal elements equal to 1,2,3,4,5.

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Transcript

00:01 In this question, we have to check whether the given matrix are diagonalizable or not.

00:06 So we have 211041, then next is 006.

00:12 Let's call this matrix as a, which is given to be an upper triangular matrix.

00:19 So the entries on the diagonal, they give the eigenvalues.

00:23 So we write down the eigenvalues of this matrix a equal to 2, 4 and 6.

00:32 All are distinct.

00:36 So this will implies that the matrix a is diagonalizable.

00:45 That is, a can be written in the form of pdp inverse where d will be consisting of the eigenvalues only along the diagonal.

01:01 This is a diagonal matrix, correct? and p is an invertible matrix which is formed with the eigenvector as its column vector.

01:15 So, we are our aim is to find out those eigenvector here.

01:21 Let us find out now the eigenvector.

01:26 First we find for lambda equals to 2.

01:30 We consider a minus 2i times x equal to 0.

01:34 A minus 2y it will be 2 minus 2 is 0 1 1 0 2 1 0 0 4 times x1 x2 x3 equal to 0.

01:50 This is what we are getting correct.

01:58 So simplifying this we get x1 x2 x3 the eigenvector as 1 0 0.

02:06 Similarly for the second eigenvalue which is lambda equals to 4, we get a minus 4i times x equal to 0...

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