## 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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