/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 57 What would a \(5 \times 5\) matr... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 57

What would a \(5 \times 5\) matrix \(A\) look like if \(A_{i i}=0\) for every \(i\) ?

Short Answer

Expert verified
The 5x5 matrix A with the condition that \(A_{i i} = 0\) for every \(i\) will be: \[ A = \begin{bmatrix} 0 & a_{12} & a_{13} & a_{14} & a_{15}\\ a_{21} & 0 & a_{23} & a_{24} & a_{25}\\ a_{31} & a_{32} & 0 & a_{34} & a_{35}\\ a_{41} & a_{42} & a_{43} & 0 & a_{45}\\ a_{51} & a_{52} & a_{53} & a_{54} & 0\\ \end{bmatrix} \] where all the other elements remain unchanged, and the diagonal elements are equal to zero.

Step by step solution

01

Understand the condition

We are given that \(A_{i i} = 0\) for every \(i\). This means that the diagonal of the matrix A consists of only zeros.
02

Write out the general form of a 5x5 matrix

A 5x5 matrix A can be written as: \[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15}\\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25}\\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35}\\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45}\\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55}\\ \end{bmatrix} \]
03

Apply the given condition

Now, we have to replace all diagonal elements by zero, as per the given condition \(A_{i i} = 0\) for every \(i\). The updated matrix A will be: \[ A = \begin{bmatrix} 0 & a_{12} & a_{13} & a_{14} & a_{15}\\ a_{21} & 0 & a_{23} & a_{24} & a_{25}\\ a_{31} & a_{32} & 0 & a_{34} & a_{35}\\ a_{41} & a_{42} & a_{43} & 0 & a_{45}\\ a_{51} & a_{52} & a_{53} & a_{54} & 0\\ \end{bmatrix} \]
04

Final solution

The 5x5 matrix A with the condition that \(A_{i i} = 0\) for every \(i\) will be: \[ A = \begin{bmatrix} 0 & a_{12} & a_{13} & a_{14} & a_{15}\\ a_{21} & 0 & a_{23} & a_{24} & a_{25}\\ a_{31} & a_{32} & 0 & a_{34} & a_{35}\\ a_{41} & a_{42} & a_{43} & 0 & a_{45}\\ a_{51} & a_{52} & a_{53} & a_{54} & 0\\ \end{bmatrix} \] where all the other elements remain unchanged, and the diagonal elements are equal to zero.

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