An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Approximately 2.81 × 10^21 electrons flow through an electric device when a current of 15.0 A is delivered for 30 seconds. This is calculated using the relationship between current, charge, and the charge of a single electron. The total charge is found to be 450 coulombs, which is then converted into the number of electrons.
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Determine the algebraic multiplicity of eigenvalue λ 1 = 4 from the characteristic polynomial: 2.
Calculate A − 4 I and find its rank through row reduction: rank is 2.
Calculate the geometric multiplicity: 4 − r ank ( A − 4 I ) = 2 .
Since algebraic multiplicity equals geometric multiplicity, there are 2 Jordan blocks of size 1x1: 2 .
Explanation
Problem Setup and Objective We are given the matrix A = 4 0 − 1 − 4 0 4 − 1 − 4 0 0 0 0 0 0 − 1 0 and its characteristic polynomial χ A ( λ ) = ( λ − 4 ) 2 λ 2 . We want to determine the size and number of Jordan blocks corresponding to the eigenvalue λ 1 = 4 .
Algebraic Multiplicity First, we find the algebraic multiplicity of the eigenvalue λ 1 = 4 . From the characteristic polynomial, the algebraic multiplicity is 2, since the exponent of ( λ − 4 ) is 2.
Calculating A - 4I Next, we find the geometric multiplicity of the eigenvalue λ 1 = 4 . This is given by the dimension of the eigenspace corresponding to λ 1 = 4 , which is d im ( k er ( A − 4 I )) . We calculate A − 4 I :
A − 4 I = 4 − 4 0 − 1 − 4 0 4 − 4 − 1 − 4 0 0 0 − 4 0 0 0 − 1 0 − 4 = 0 0 − 1 − 4 0 0 − 1 − 4 0 0 − 4 0 0 0 − 1 − 4
Finding the Rank of A - 4I Now we find the rank of A − 4 I . We can row reduce the matrix to find its rank. The matrix is A − 4 I = 0 0 − 1 − 4 0 0 − 1 − 4 0 0 − 4 0 0 0 − 1 − 4 We can swap rows 1 and 3, and then multiply by -1: 1 0 0 − 4 1 0 0 − 4 4 0 0 0 1 0 0 − 4 Add 4 times row 1 to row 4: 1 0 0 0 1 0 0 0 4 0 0 16 1 0 0 0 Swap rows 2 and 4: 1 0 0 0 1 0 0 0 4 16 0 0 1 0 0 0 Divide row 2 by 16: 1 0 0 0 1 0 0 0 4 1 0 0 1 0 0 0 Subtract 4 times row 2 from row 1: 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 The rank of this matrix is 2.
Geometric Multiplicity The geometric multiplicity is 4 − r ank ( A − 4 I ) = 4 − 2 = 2 .
Jordan Block Structure Since the algebraic multiplicity is 2 and the geometric multiplicity is 2, there are 2 Jordan blocks of size 1x1.
Final Answer Therefore, the Jordan block corresponding to the eigenvalue λ 1 = 4 consists of 2 Jordan blocks of size 1x1.
Examples
Jordan Normal Form is used in linear algebra to simplify matrix calculations, especially when dealing with eigenvalues and eigenvectors. For example, in physics, it can be used to analyze the stability of a system near an equilibrium point. The matrix A represents a linear transformation, and understanding its Jordan form helps in predicting the system's behavior over time. This is particularly useful in fields like control theory and quantum mechanics, where matrix diagonalization simplifies complex calculations.
