## Linear Algebra and Its Applications, Review Exercise 1.25

Review exercise 1.25. Given the matrix $A$ where $A = \begin{bmatrix} 1&0&0 \\ 2&1&0 \\ 0&5&1 \end{bmatrix} \begin{bmatrix} 1&2&0 \\ 0&1&5 \\ 0&0&1 \end{bmatrix}$

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

Answer: From the above we see that $A$ has been factored into $A = LL^T$. Since $L$ contains the multipliers used in elimination, we can look at $l_32 = 5$ to determine that 5 times row 2 was subtracted from row 3 in elimination.

Since $A$ can be factored into the product of a lower triangular matrix $L$ and upper triangular matrix $L^T$ we know that $A$ is invertible. Since $A = LL^T$ we have $A^T = (LL^T)^T = (L^T)^TL^T = LL^T = A$ so $A$ is also symmetrical. Since $L$ and $L^T$ are bidiagonal the original matrix $A$ was tridiagonal. Finally, since we can express $A$ as $A = LDL^T$ where $D = I$ 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 .

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