Question
Let $A$ be an $n \times n$ nonzero matrix satisfying $A^{10}=O .$ Explain why $A$ must be singular. What properties of determinants are you using in your argument?
Step 1
Step 1: We are given that $A^{10}=O$, where $A$ is a non-zero $n \times n$ matrix and $O$ is the zero matrix. Show more…
Show all steps
Your feedback will help us improve your experience
Vaibhav Jain and 68 other Algebra educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Suppose $A$ is an $n \times n$ matrix with the property that the equation $A \mathbf{x}=\mathbf{0}$ has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation $A \mathbf{x}=\mathbf{b}$ must have a solution for each $\mathbf{b}$ in $\mathbb{R}^{n}$ .
Matrix Algebra
Characterizations of Invertible Matrices
Let $A$ be an $m \times n$ matrix of rank $r>0$ and let $U$ be an echelon form of $A .$ Explain why there exists an invertible matrix $E$ such that $A=E U,$ and use this factorization to write $A$ as the sum of $r$ rank 1 matrices. [Hint: See Theorem 10 in Section $2.4 . ]$
Vector Spaces
Rank
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD