Linear Algebra. **Problem Statement**Let $ A $ be an $ n \times n $ matrix such that $ A^m = 0 $ for some positive integer $ m $. Show that $ A $ is not invertible. Give a full proof.**Explanation**This problem concerns a square matrix $ A $ and requires proving that it is not invertible (also known as singular) if there exists some positive integer $ m $ such that raising $ A $ to the $ m $-th power results in the zero matrix. This typically involves demonstrating that the existence of this power implies certain properties of $ A $, such as the lack of a full set of linearly independent columns or rows, thereby having a determinant of zero. To solve this problem, you will have to rely on linear algebra concepts like matrix powers, determinants, and the definition of invertibility.

A is a n×n invertible matrix, such that Am=0⋯⋯1 Taking determinants both side, detAm=det0detAm=0…

Answered: Let A be an n x n matrix such that Am =…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Linear Algebra.

**Problem Statement**

Let $ A $ be an $ n \times n $ matrix such that $ A^m = 0 $ for some positive integer $ m $. Show that $ A $ is not invertible. Give a full proof.

**Explanation**

This problem concerns a square matrix $ A $ and requires proving that it is not invertible (also known as singular) if there exists some positive integer $ m $ such that raising $ A $ to the $ m $-th power results in the zero matrix. This typically involves demonstrating that the existence of this power implies certain properties of $ A $, such as the lack of a full set of linearly independent columns or rows, thereby having a determinant of zero.

To solve this problem, you will have to rely on linear algebra concepts like matrix powers, determinants, and the definition of invertibility.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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