(a) Let A be an m ×n matrix and let M=(A |𝐛) be the augmented matrix for the linear system A 𝐱=𝐛

Name: Problem 16, Chapter 1: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2022-02-26T04:57:11-08:00
Duration: 6 min 35 s
Channel: Uma Kumari
Description: (a) Let A be an m ×n matrix and let M=(A |𝐛) be the augmented matrix for the linear system A 𝐱=𝐛. Show that either (i) rank A=rank M, or (ii) rank A=rank M-1. (b) Prove that the system is compatible if and only if case (i) holds.

step 1 In this question we have a matrix A the order of this matrix is m into n we are required to prov

Question

(a) Let $A$ be an $m \times n$ matrix and let $M=(A \mid \mathbf{b})$ be the augmented matrix for the linear system $A \mathbf{x}=\mathbf{b}$. Show that either (i) $\operatorname{rank} A=\operatorname{rank} M$, or (ii) $\operatorname{rank} A=\operatorname{rank} M-1$. (b) Prove that the system is compatible if and only if case (i) holds.

    (a) Let $A$ be an $m \times n$ matrix and let $M=(A \mid \mathbf{b})$ be the augmented matrix for the linear system $A \mathbf{x}=\mathbf{b}$. Show that either (i) $\operatorname{rank} A=\operatorname{rank} M$, or (ii) $\operatorname{rank} A=\operatorname{rank} M-1$.
(b) Prove that the system is compatible if and only if case (i) holds.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 1, Problem 16

Instant Answer

Solved by Expert Uma Kumari

02/26/2022

Step 1

- The matrix $A$ is an $m \times n$ matrix. - The vector $\mathbf{b}$ is an $m \times 1$ column vector. - The augmented matrix $M = (A \mid \mathbf{b})$ is formed by appending the column vector $\mathbf{b}$ to the matrix $A$, resulting in an $m \times (n+1)$ Show more…

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(a) Let $A$ be an $m \times n$ matrix and let $M=(A \mid \mathbf{b})$ be the augmented matrix for the linear system $A \mathbf{x}=\mathbf{b}$. Show that either (i) $\operatorname{rank} A=\operatorname{rank} M$, or (ii) $\operatorname{rank} A=\operatorname{rank} M-1$. (b) Prove that the system is compatible if and only if case (i) holds.

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Key Concepts

Matrix Rank

Matrix rank is a fundamental concept that indicates the maximum number of linearly independent columns (or rows) in a matrix. It reflects the dimension of the column space and is crucial for understanding the behavior of linear transformations and systems of equations.

Augmented Matrix

An augmented matrix is formed by appending the column of constants from a system of linear equations to the coefficient matrix. This construction is used to analyze the system’s consistency by comparing the rank of the coefficient matrix with the rank of the augmented matrix.

Consistency of Linear Systems

The consistency of a linear system refers to whether the system has at least one solution. A system is consistent if the vector of constants lies within the column space of the coefficient matrix, which is characterized by the condition that the rank of the coefficient matrix equals the rank of the augmented matrix.

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