Linear Algebra and Its Applications, Exercise 3.1.4

Exercise 3.1.4. If B is an invertible matrix, describe why row i of B and column j of B^{-1} are orthogonal in the case i \ne j.

Answer: We have BB^{-1} = I. The identity matrix I has ones on the diagonal (i.e., when i = j) and zeros otherwise (when i \ne j).

By the rules of matrix multiplication the ij entry of the matrix product BB^{-1} is equal to the inner product of row i of B with column j of B^{-1}. Since BB^{-1} = I that inner product must be zero when i \ne j, so that row i of B and column j of B^{-1} 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.

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