Let A be the n ×n matrix with all 1 's on the diagonal and all 0^∘ s above and below the diagona

Name: Problem 33, Chapter 1: Linear Algebra With Applications
Uploaded: 2023-01-29T03:56:35-08:00
Channel: Nick Johnson
Description: Let A be the n ×n matrix with all 1 's on the diagonal and all 0^∘ s above and below the diagonal. What is A x⃗ where x⃗ is a vector in ℝ^n ?

step 1 Hello, so here we're going to let A be an N by N matrix where we have all ones on the diagonal a

Question

Let $A$ be the $n \times n$ matrix with all 1 's on the diagonal and all $0^{\circ}$ s above and below the diagonal. What is $A \vec{x}$ where $\vec{x}$ is a vector in $\mathbb{R}^{n}$ ?

   Let $A$ be the $n \times n$ matrix with all 1 's on the diagonal and all $0^{\circ}$ s above and below the diagonal. What is $A \vec{x}$ where $\vec{x}$ is a vector in $\mathbb{R}^{n}$ ?

Linear Algebra With Applications

Otto Bretscher 5th Edition

Chapter 1, Problem 33

Instant Answer

Solved by Expert Nick Johnson

01/29/2023

Step 1

Step 1: First, we define the matrix $A$ as an $n \times n$ matrix where the elements $a_{ij}$ are defined as follows: \[a_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}\] This means that $A$ is a diagonal matrix with all ones on Show more…

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Let $A$ be the $n \times n$ matrix with all 1 's on the diagonal and all $0^{\circ}$ s above and below the diagonal. What is $A \vec{x}$ where $\vec{x}$ is a vector in $\mathbb{R}^{n}$ ?

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Key Concepts

Diagonal Matrix

A diagonal matrix is one where all of its non-diagonal entries are zero. Multiplying a diagonal matrix by a vector results in each component of the vector being scaled by the corresponding diagonal element. In this problem, the matrix only has nonzero entries on the diagonal, making the multiplication straightforward.

Identity Matrix

An identity matrix is a special kind of diagonal matrix where every entry on the diagonal is one. Multiplying any vector by the identity matrix leaves the vector unchanged, as each component of the vector is multiplied by one.

Matrix-Vector Multiplication

Matrix-vector multiplication involves taking the dot product of each row of the matrix with the vector. When the matrix is the identity matrix (with ones on the diagonal and zeros elsewhere), this operation results in each component of the vector being preserved, effectively leaving the vector unchanged.

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