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.

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