## Linear Algebra and Its Applications, exercise 1.4.8

Exercise 1.4.8. Give non-zero examples of 3×3 matrices with the following properties:

1. diagonal matrix ( $a_{ij} = 0$ if $i \neq j$);
2. symmetric matrix ( $a_{ij} = a_{ji}$ for all $i \neq j$);
3. upper triangular matrix ( $a_{ij} = 0$ if $i > j$);
4. skew-symmetric matrix ( $a_{ij} = -a_{ji}$ for all i and j). $\begin{bmatrix} 1&0&0 \\ 0&2&0 \\ 0&0&3 \end{bmatrix}$

Symmetric matrix: $\begin{bmatrix} 1&2&3 \\ 2&2&4 \\ 3&4&3 \end{bmatrix}$

Upper triangular matrix: $\begin{bmatrix} 1&2&3 \\ 0&2&3 \\ 0&0&3 \end{bmatrix}$

Skew-symmetric matrix: $\begin{bmatrix} 0&2&3 \\ -2&0&4 \\ -3&-4&0 \end{bmatrix}$

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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