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How do we know that the ith row of an invertible matrix B is orthogonal to the jth column of B if i j?

How do we know that the ith row of an invertible matrix B is orthogonal to the jth column of B if i j?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Understanding Orthogonality in Invertible Matrices** *How do we know that the \(i\)th row of an invertible matrix \(B\) is orthogonal to the \(j\)th column of \(B^{-1}\), if \(i \neq j\)?* This question explores a fundamental property of invertible matrices, particularly focusing on the concept of orthogonality between specific rows and columns. In the context of linear algebra, understanding the relationship between a matrix and its inverse is crucial, as it highlights important characteristics about linear transformations and vector spaces. If row \(i\) of matrix \(B\) and column \(j\) of \(B^{-1}\) are orthogonal when \(i \neq j\), it suggests that their dot product equals zero, reflecting an intrinsic geometric property of the matrix inverse.
Transcribed Image Text:**Understanding Orthogonality in Invertible Matrices** *How do we know that the \(i\)th row of an invertible matrix \(B\) is orthogonal to the \(j\)th column of \(B^{-1}\), if \(i \neq j\)?* This question explores a fundamental property of invertible matrices, particularly focusing on the concept of orthogonality between specific rows and columns. In the context of linear algebra, understanding the relationship between a matrix and its inverse is crucial, as it highlights important characteristics about linear transformations and vector spaces. If row \(i\) of matrix \(B\) and column \(j\) of \(B^{-1}\) are orthogonal when \(i \neq j\), it suggests that their dot product equals zero, reflecting an intrinsic geometric property of the matrix inverse.
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