Review exercise 1.25. Given the matrix where

what multiple of row 2 was subtracted from row 3 in elimination? Explain why is invertible, symmetric, and tridiagonal. What are the pivots?

Answer: From the above we see that has been factored into . Since contains the multipliers used in elimination, we can look at to determine that 5 times row 2 was subtracted from row 3 in elimination.

Since can be factored into the product of a lower triangular matrix and upper triangular matrix we know that is invertible. Since we have so is also symmetrical. Since and are bidiagonal the original matrix was tridiagonal. Finally, since we can express as where we see that the pivots (i.e., the diagonal entries of D) are all 1.

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.