Exercise 3.1.4. If is an invertible matrix, describe why row of and column of are orthogonal in the case .
Answer: We have . The identity matrix has ones on the diagonal (i.e., when ) and zeros otherwise (when ).
By the rules of matrix multiplication the entry of the matrix product is equal to the inner product of row of with column of . Since that inner product must be zero when , so that row of and column of are orthogonal vectors in that case.
NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.
If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.