1. For a square matrix A, if Ax = λx for some vector x ≠ 0, then λ is called:
Correct Answer: b) Eigenvalue of A
Explanation: λ is called an eigenvalue of a square matrix A if there exists a non-zero vector x such that Ax = λx.
2. The non-zero vector x in the equation Ax = λx is called:
Correct Answer: c) Either a or b
Explanation: The non-zero vector x in the equation Ax = λx is called an eigenvector of the matrix A, corresponding to the eigenvalue λ.
3. The characteristic equation of a matrix A is given by:
Correct Answer: b) det(A - λI) = 0
Explanation: The characteristic equation of a matrix A is given by det(A - λI) = 0, where λ represents the eigenvalues of A and I is the identity matrix of the same dimension as A.
4. The eigenvalues of a diagonal matrix are:
Correct Answer: a) Its diagonal elements
Explanation: The eigenvalues of a diagonal matrix are simply its diagonal elements since multiplying a diagonal matrix by a vector results in a scaled version of that vector, with the scaling factor being the corresponding diagonal element.
5. The sum of eigenvalues of a matrix equals its:
Correct Answer: b) Trace
Explanation: The sum of the eigenvalues of a square matrix is equal to its trace, which is the sum of its diagonal elements. This property holds true for all square matrices.
6. The product of eigenvalues of a matrix equals its:
Correct Answer: b) Determinant
Explanation: The product of the eigenvalues of a square matrix is equal to its determinant. This is a fundamental property of matrices in linear algebra.
7. The Cayley-Hamilton theorem states that:
Correct Answer: a) Every matrix satisfies its characteristic equation
Explanation: The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation, meaning if the characteristic polynomial of a matrix A is p(λ) = det(A - λI), then substituting A into its polynomial results in p(A) = 0.
8. If A is a 2×2 matrix with characteristic equation λ² - 5λ + 6 = 0, then A² equals:
Correct Answer: a) 5A - 6I
Explanation: According to the Cayley-Hamilton theorem, every square matrix satisfies its own characteristic equation. For the matrix A, the characteristic equation is λ² - 5λ + 6 = 0. Substituting A into this equation gives A² - 5A + 6I = 0 ⟹ A² = 5A - 6I.
9. If λ is an eigenvalue of an invertible matrix A, then an eigenvalue of A⁻¹ is:
Correct Answer: b) 1/λ
Explanation: If λ is an eigenvalue of an invertible matrix A, then an eigenvalue of A⁻¹ (the inverse of A) is 1/λ. This follows from the property A⁻¹x = (1/λ)x, where x is the corresponding eigenvector.
10. If a matrix has distinct eigenvalues, then:
Correct Answer: b) It is always diagonalizable
Explanation: A matrix with distinct eigenvalues is always diagonalizable. This is because distinct eigenvalues guarantee the existence of linearly independent eigenvectors, which can form a basis for diagonalization.