Linear Algebra | Senioritis

John Rhodes
July 30, 2023
Linear Algebra

Proving the Positive Semidefiniteness of AA^T | Mathematical Analysis of Eigenvalues and Determinants.

. For all n × n matrices A, det(AA^T ) ≥ 0. (T/F) The statement is true The statement is true. To prove this, let’s begin by...

John Rhodes
July 30, 2023
Linear Algebra

Understanding Matrix Invertibility | Can the Product of Non-Square Matrices A and B be Invertible?

There exists a 2 × 3 matrix A and a 3 × 2 matrix B such that the product AB is invertible. (T/F) False False. In order...

John Rhodes
July 30, 2023
Linear Algebra

Understanding Eigenvalues | The Truth about Real Eigenvalues for 2 × 2 Matrices

Every 2 × 2 matrix has at least one real eigenvalue. (T/F) The statement “Every 2 × 2 matrix has at least one real eigenvalue” is true...

John Rhodes
July 30, 2023
Linear Algebra

Proving that a Symmetric 2×2 Matrix Always Has Two Real Eigenvalues

Every symmetric (that is, A = A^T ) 2 × 2 matrix always has two real eigenvalues, counting multiplicities. (T/F) True True. A symmetric 2×2 matrix will...

John Rhodes
July 30, 2023
Linear Algebra

Understanding Matrix Similarity | Debunking the False Statement If A is similar to λI, then A = λI

If A is similar to λI for some scalar λ, then A = λI. (T/F) The statement “If A is similar to λI for some scalar λ,...

John Rhodes
July 30, 2023
Linear Algebra

Understanding the Reversibility of Elementary Row Operations in Linear Algebra

Every elementary row operation is reversible In linear algebra, elementary row operations refer to a set of three operations that can be performed on the rows of...

John Rhodes
July 30, 2023
Linear Algebra

Understanding the Solution Set of a Linear System | Explained with Examples

The solution set of a linear system involving variables x1, …, xn is a list of numbers (s1,…,sn) that makes each equation in the system a true...

John Rhodes
July 30, 2023
Linear Algebra

The Importance of Existence and Uniqueness in Linear Systems | A Comprehensive Guide

Two fundamental questions about a linear system involve existence and uniqueness. Existence and uniqueness are important concepts when studying linear systems in mathematics Existence and uniqueness are...