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What is the rank of a matrix?
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The rank of a matrix is the maximum number of linearly independent row vectors in the matrix.
Describe the characteristics of eigenvalues for real symmetric matrices.
Eigenvalues of real symmetric matrices are always real.
What is the sum of the eigenvalues of a matrix?
The sum of the eigenvalues of a matrix equals the trace of the matrix.
Define an involutory matrix and provide an example.
A matrix A is involutory if A^2 = I. Example: A = [4 -1] [15 -4]
How is the characteristic equation of a matrix formed to determine its eigenvalues?
The characteristic equation is formed as det(A - λI) = 0.
Explain the concept of a matrix being idempotent.
A matrix A is idempotent if A^2 = A.
What does it mean for a matrix to be symmetric?
A matrix is symmetric if it equals its transpose, A = A^T.
Define an orthogonal matrix and provide an example.
A matrix A is orthogonal if A * A^T = I. Example: A = [cosθ -sinθ 0] [sinθ cosθ 0] [0 0 1]
Define a matrix and give an example.
A matrix is an arrangement of elements in rows and columns. Example: A = [i n d i a] [j a p a n]
What is the determinant of an upper triangular matrix?
The determinant of an upper triangular matrix is the product of its diagonal elements.
Explain the condition required for matrix multiplication.
Two matrices A (MxN) and B (PxQ) can be multiplied if N = P. The resultant matrix will be of size MxQ.
What defines a scalar matrix, and what are its special cases?
A scalar matrix is a diagonal matrix where all principal diagonal elements are equal. Special cases: Identity matrix (k=1) and null matrix (k=0).
What distinguishes a non-homogeneous system from a homogeneous system in matrix form?
A non-homogeneous system is in the form A * X = B, while a homogeneous system is in the form A * X = 0.
What property of matrix multiplication describes (A * B) * C = A * (B * C)?
This describes the associative property of matrix multiplication.
When is a matrix said to be nilpotent?
A matrix A is nilpotent if there exists some positive integer k such that A^k = 0.
How do you determine if a system of equations has a unique solution?
A system of equations has a unique solution if rank(A) = rank(A|B) = number of unknowns.
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