MCQ Subject: Linear Algebra & Differential Equations

In Linear Algebra, what geometric or algebraic property of a square matrix is directly represented by the sum of all its eigenvalues?

The stability of a linear system of differential equations can be determined by analyzing the eigenvalues of its system matrix. The determinant of this matrix, which is crucial for stability analysis, is equal to the _______ of its eigenvalues.

In the context of a matrix A, what do eigenvalues (λ) primarily represent when Av = λv?

A necessary and sufficient condition for an n×n matrix to be diagonalizable is that it has

In analyzing the stability of a linear system of differential equations, what primarily determines the behavior of the system’s solutions?

The sum of all eigenvalues of a square matrix corresponds to which of the following matrix properties?

What geometric interpretation best describes the effect of an eigenvalue λ on its corresponding eigenvector in a linear transformation?

For a square matrix A with eigenvalues λ₁, λ₂, …, λₙ, which of the following statements is TRUE?

The characteristic equation of a second-order linear homogeneous differential equation has roots -1 and -2. As t approaches infinity, what happens to the solution of the DE?

A system with state matrix A has eigenvalues λ = -1, λ = -2 ± j3. What is the stability of the system?