Linear Algebra Concepts Quiz

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Questions and Answers

What is the rank of a matrix determined by elementary transformations?

The rank becomes zero
The rank remains unchanged (correct)
The rank increases by 1
The rank decreases by 1

In the context of matrices, what is the Cayley-Hamilton Theorem related to?

Eigenvalues and eigenvectors (correct)
System of linear equations
Symmetric matrices
Orthogonal matrices

What type of matrix satisfies the property A^T = -A?

Skew symmetric matrix (correct)
Orthogonal matrix
Symmetric matrix
Diagonal 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. Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

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