6 Questions
What is the rank of a matrix determined by elementary transformations?
The rank remains unchanged
In the context of matrices, what is the Cayley-Hamilton Theorem related to?
Eigenvalues and eigenvectors
What type of matrix satisfies the property A^T = -A?
Skew symmetric matrix
Define eigenvalues and eigenvectors of a matrix.
Eigenvalues and eigenvectors of a matrix are values and vectors that satisfy the equation Av = λv, where A is a square matrix, v is a non-zero vector, and λ is a scalar. The eigenvectors are the non-zero vectors that satisfy this equation, and the eigenvalues are the corresponding scalars.
Explain the Cayley-Hamilton Theorem in the context of matrices.
The Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic equation. In other words, if A is a square matrix, then the characteristic polynomial of A, denoted by p(λ), when evaluated at A, results in the zero matrix: p(A) = 0.
What are symmetric, skew-symmetric, and orthogonal matrices?
Symmetric matrices are square matrices that are equal to their transpose: A = A^T. Skew-symmetric matrices are square matrices that satisfy the property A^T = -A. Orthogonal matrices are square matrices that satisfy the property A^T * A = A * A^T = I, where I is the identity matrix.
Test your knowledge of matrices, rank of a matrix, systems of linear equations, symmetric, skew symmetric, and orthogonal matrices, eigenvalues, eigenvectors, and the Cayley-Hamilton Theorem with this quiz.
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