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What is the characteristic polynomial of matrix A if A = [1 2; 3 4]?
What is the characteristic polynomial of matrix A if A = [1 2; 3 4]?
Which of the following expressions correctly represents the characteristic equation derived from the characteristic polynomial λ² - 4λ - 5?
Which of the following expressions correctly represents the characteristic equation derived from the characteristic polynomial λ² - 4λ - 5?
If matrix A has eigenvalues λ = 5, -3, 7/3, what is the sum of the eigenvalues?
If matrix A has eigenvalues λ = 5, -3, 7/3, what is the sum of the eigenvalues?
For the matrix B = 5A⁴ - 7A³ + 2A - 3A¹, what is the method to determine the eigenvalues of B based on the eigenvalues of A?
For the matrix B = 5A⁴ - 7A³ + 2A - 3A¹, what is the method to determine the eigenvalues of B based on the eigenvalues of A?
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What is the determinant of the matrix A - λI if A = [2 0; 2 2]?
What is the determinant of the matrix A - λI if A = [2 0; 2 2]?
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What is the characteristic equation for a 3x3 matrix A?
What is the characteristic equation for a 3x3 matrix A?
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If the eigenvalues of matrix A are all equal to 2, which of the following correctly describes A?
If the eigenvalues of matrix A are all equal to 2, which of the following correctly describes A?
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What property relates the eigenvalues of the adjugate matrix Adj A to those of matrix A?
What property relates the eigenvalues of the adjugate matrix Adj A to those of matrix A?
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For a lower triangular matrix, how are the eigenvalues determined?
For a lower triangular matrix, how are the eigenvalues determined?
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Given that the determinant of matrix A is 4, which of the following statements about eigenvalues is true?
Given that the determinant of matrix A is 4, which of the following statements about eigenvalues is true?
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What condition must hold for a matrix A to be diagonalizable?
What condition must hold for a matrix A to be diagonalizable?
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Given the eigenvalue λ = 5, which of the following represents an eigenvector for matrix A if x = t [1]?
Given the eigenvalue λ = 5, which of the following represents an eigenvector for matrix A if x = t [1]?
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For which value of λ does the matrix A have an algebraic multiplicity of 2 and a geometric multiplicity of 2?
For which value of λ does the matrix A have an algebraic multiplicity of 2 and a geometric multiplicity of 2?
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What does the equation D = P⁻¹AP signify in relation to a diagonalizable matrix?
What does the equation D = P⁻¹AP signify in relation to a diagonalizable matrix?
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In the example provided, which eigenvalue has a corresponding eigenvector expressed as t(1, 1) for matrix A?
In the example provided, which eigenvalue has a corresponding eigenvector expressed as t(1, 1) for matrix A?
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What is the characteristic polynomial for the matrix A in the last example?
What is the characteristic polynomial for the matrix A in the last example?
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Under what condition is the matrix A considered not diagonalizable in the last example?
Under what condition is the matrix A considered not diagonalizable in the last example?
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What is the relationship between x₁, x₂, and x₃ in the example with λ = 4?
What is the relationship between x₁, x₂, and x₃ in the example with λ = 4?
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What does the Cayley Hamilton Theorem state regarding matrix A?
What does the Cayley Hamilton Theorem state regarding matrix A?
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Which term correctly represents the eigenvalue λ associated with matrix A in the example if λ = 9 and is repeated thrice?
Which term correctly represents the eigenvalue λ associated with matrix A in the example if λ = 9 and is repeated thrice?
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If the equation $ ext{sin } A = pA + qI$ is valid, what can be inferred about the relationship of $p$ and $q$ when $ ext{sin } π = 0$?
If the equation $ ext{sin } A = pA + qI$ is valid, what can be inferred about the relationship of $p$ and $q$ when $ ext{sin } π = 0$?
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What is the result of simplifying $e^{1t} + e^{-1t} - e^{1t} - e^{-1t} + 1 = 0$?
What is the result of simplifying $e^{1t} + e^{-1t} - e^{1t} - e^{-1t} + 1 = 0$?
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For the function $ ext{sin } λ = pλ + q$, what is the value of $p$ when $λ = 3π/2$?
For the function $ ext{sin } λ = pλ + q$, what is the value of $p$ when $λ = 3π/2$?
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What roots does the cubic polynomial $λ^3 - 6λ^2 + 9λ - 4 = 0$ yield?
What roots does the cubic polynomial $λ^3 - 6λ^2 + 9λ - 4 = 0$ yield?
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Considering the equation $λ^2 + 1 = 0$, what values does $λ$ take?
Considering the equation $λ^2 + 1 = 0$, what values does $λ$ take?
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What is the significance of the formula $e^{tA} = pA + qI$ in terms of matrix exponentials?
What is the significance of the formula $e^{tA} = pA + qI$ in terms of matrix exponentials?
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If $A = egin{bmatrix} 3 & 4 & -1 \ 5 & -2 & 1 \ 2 & 1 & 3 \ ext{ and } A^{-1} = rac{1}{5} egin{bmatrix} 2 & 4 & -1 \ 1 & 3 & -1 \ -1 & -1 & 2 \ ext{, what is the relationship of the matrices } A ext{ and } A^{-1}?$
If $A = egin{bmatrix} 3 & 4 & -1 \ 5 & -2 & 1 \ 2 & 1 & 3 \ ext{ and } A^{-1} = rac{1}{5} egin{bmatrix} 2 & 4 & -1 \ 1 & 3 & -1 \ -1 & -1 & 2 \ ext{, what is the relationship of the matrices } A ext{ and } A^{-1}?$
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When solving for $q$ from the equation $3π/2 + q = -1$, what is the correct value?
When solving for $q$ from the equation $3π/2 + q = -1$, what is the correct value?
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Study Notes
Eigenvalues and Eigenvectors
- The characteristic polynomial of a square matrix A is defined as
f(λ) = det(A - λI), whereλis an eigenvalue andIis the identity matrix. - The characteristic equation is found by setting the characteristic polynomial equal to zero:
det(A - λI) = 0. - The roots of the characteristic equation are the eigenvalues of the matrix.
- To calculate the eigenvalues of a matrix, find the determinant of
(A - λI), set the determinant equal to zero, and solve the resulting equation forλ. - The eigenvalues of a matrix can be used to determine the stability of a system.
Properties of Eigenvalues
- The sum of the eigenvalues of a matrix is equal to the trace of the matrix.
- The product of the eigenvalues of a matrix is equal to the determinant of the matrix.
- For a 2x2 matrix, the characteristic equation is
λ² - (trace A)λ + det A = 0. - For a 3x3 matrix, the characteristic equation is
λ³ - (trace A)λ² + (c₁₁ + c₂₂ + c₃₃)λ - det A = 0, wherecᵢⱼare the elements of the matrix. - If A is a diagonal matrix, an upper triangular matrix, or a lower triangular matrix, then the diagonal elements are the eigenvalues of A.
- If
λis an eigenvalue of A, theng(λ)is an eigenvalue ofg(A), whereg(x)is any polynomial. - The eigenvalues of
Adj Aaredet(A)/λ.
Eigenvectors
- An eigenvector of a matrix is a non-zero vector that, when multiplied by the matrix, is simply scaled by a factor equal to the corresponding eigenvalue.
- To find the eigenvectors of a matrix, solve the equation
(A - λI)x = 0, whereλis an eigenvalue andxis the eigenvector.
Diagonalizable Matrices
- A matrix is said to be diagonalizable if it can be transformed into a diagonal matrix by a similarity transformation.
- A matrix is diagonalizable if and only if the algebraic multiplicity of each eigenvalue is equal to its geometric multiplicity.
- If a matrix A is diagonalizable, there exists a diagonal matrix D and a diagonalizing matrix P such that
D = P⁻¹AP. - A matrix with distinct eigenvalues is always diagonalizable.
Examples
-
Example 1:
A = [ 1 4 -2 ][ -2 4 -2 ][ -2 4 2 ] -
For this matrix, the characteristic equation is
λ³ - 4λ² + 6λ - 6 = 0. -
The algebraic multiplicity of the eigenvalue 9 is 2, and the geometric multiplicity is also 2, meaning the matrix is diagonalizable.
-
The diagonal matrix D is
[9 0 0].[ 0 18 0][ 0 0 18] -
The diagonalizing matrix P is
[ -2 -1 1 ].[ 0 1 2 ][ 1 2 0 ] -
Example 2:
A = [ 1 3 7 ][ 4 2 3 ][ 1 1 2 ] -
The characteristic equation is
λ³ - 4λ² - 20λ - 35 = 0. -
The algebraic multiplicity of each eigenvalue is 1, but the geometric multiplicity is only 1.
-
This matrix is not diagonalizable.
Cayley-Hamilton Theorem
- The Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic equation.
- This theorem can be used to find the inverse of a matrix, calculate powers of a matrix, and solve systems of differential equations.
Application: Finding the Sine of a Matrix
- If A is an upper triangular matrix, then
sin A = pA + qI, wherepandqare constants. - To find
pandq, we can use the following:-
sin λ = pλ + q - Substitute specific values of
λto solve forpandq.
-
Application: Exponential of a Matrix
- The exponential of a matrix is defined as
eᵗᴬ = pA + qI, wherepandqare constants. - To find
pandq, we can use the following:-
e¹ᵗ = pλ + q - Use properties of exponential functions and substitute appropriate values of
λandt.
-
Application: Inverse of a Matrix
- The inverse of a matrix can be found using the formula
A⁻¹ = Adj(A) / det(A), whereAdj(A)is the adjugate of A.
Key Examples
- The text includes examples of calculating eigenvalues and eigenvectors, and determining diagonalizability.
- These examples demonstrate the core concepts and provide practical applications for understanding the theory.
- The text also includes examples of finding the inverse of a matrix and calculating the sine and exponential of a matrix.
- These examples highlight important applications of eigenvalues and eigenvectors in linear algebra.
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Description
Test your understanding of eigenvalues and eigenvectors in linear algebra. This quiz covers the characteristic polynomial, eigenvalue calculation, and properties of eigenvalues such as trace and determinant relationships. Sharpen your skills and ensure you're ready for advanced concepts!
