## Linear Algebra and Its Applications, Exercise 2.3.20

Exercise 2.3.20.Consider the set of all 2 by 2 matrices that have the sum of their rows equal to the sum of their columns. What is a basis for this subspace? Consider the analogous set of 3 by 3 matrices with equal row and column sums. List five linearly independent matrices from this set.

Answer: Any 2 by 2 matrix $A$ in the set will have the form $A = \begin{bmatrix} a&b \\ b&a \end{bmatrix}$

with the sum of every row and every column being $a+b$. Any such matrix $A$ can be represented as a linear combination of two matrices as follows: $A = a \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix} + b \begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix}$

Since the two matrices are linearly independent and span the subspace they are a basis for the subspace.

The following five matrices are linearly independent members of the analogous set for 3 by 3 matrices: $\begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix} \quad \begin{bmatrix} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{bmatrix} \quad \begin{bmatrix} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{bmatrix}$ $\begin{bmatrix} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{bmatrix} \quad \begin{bmatrix} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{bmatrix}$

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 .

This entry was posted in linear algebra. Bookmark the permalink.