## Linear Algebra and Its Applications, Exercise 1.7.1

Exercise 1.7.1. Find the LDU factorization for the matrix $A = \begin{bmatrix} 1&-1&&& \\ -1&2&-1&& \\ &-1&2&-1& \\ &&-1&2&-1 \\ &&&-1&2 \end{bmatrix}$

(This is the same matrix as in equation (6) of section 1.7, only with $a_{11} = 1$ instead of $a_{11} = 2$.)

Answer: For the first step of elimination we multiply the first row times the multiplier $l_{21} = -1$ and subtract it from the second row: $\begin{bmatrix} 1&-1&&& \\ -1&2&-1&& \\ &-1&2&-1& \\ &&-1&2&-1 \\ &&&-1&2 \end{bmatrix} \rightarrow \begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &-1&2&-1& \\ &&-1&2&-1 \\ &&&-1&2 \end{bmatrix}$

In the second step we multiply the second row by the multiplier $l_{32} = -1$ and subtract it from the third row: $\begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &-1&2&-1& \\ &&-1&2&-1 \\ &&&-1&2 \end{bmatrix} \rightarrow \begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &&1&-1& \\ &&-1&2&-1 \\ &&&-1&2 \end{bmatrix}$

In the third step we multiply the third row by the multiplier $l_{43} = -1$ and subtract it from the fourth row: $\begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &&1&-1& \\ &&-1&2&-1 \\ &&&-1&2 \end{bmatrix} \rightarrow \begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &&1&-1& \\ &&&1&-1 \\ &&&-1&2 \end{bmatrix}$

Finally we we multiply the fourth row by the multiplier $l_{54} = -1$ and subtract it from the fifth row: $\begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &&1&-1& \\ &&&1&-1 \\ &&&-1&2 \end{bmatrix} \rightarrow \begin{bmatrix} 1&-1&&& \\ &1&-1&& \\ &&1&-1& \\ &&&1&-1 \\ &&&&1 \end{bmatrix}$

Note that all steps feature the same multiplier. The final factorization is $A = LDU = \begin{bmatrix} &&&& \\ -1&&&& \\ &-1&&& \\ &&-1&& \\ &&&-1& \end{bmatrix} \begin{bmatrix} 1&&&& \\ &1&&& \\ &&1&& \\ &&&1& \\ &&&&1 \end{bmatrix} \begin{bmatrix} &-1&&& \\ &&-1&& \\ &&&-1& \\ &&&&-1 \\ &&&& \end{bmatrix}$

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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