Linear Algebra and Its Applications, exercise 1.4.17

Exercise 1.4.17. Assume A is a 2×2 matrix

A = \begin{bmatrix} a&b \\ c&d \end{bmatrix}

and further assume that AB = BA for any 2×2 matrix B, including the matrices

B_1 = \begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix} \quad B_2 = \begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix}

Show that a = d and that b and c are zero.

Answer: We have

AB_1 = \begin{bmatrix} a&b \\ c&d \end{bmatrix} \begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix} = \begin{bmatrix} a&0 \\ c&0 \end{bmatrix}

and

B_1A = \begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix} \begin{bmatrix} a&b \\ c&d \end{bmatrix} = \begin{bmatrix} a&b \\ 0&0 \end{bmatrix}

Since AB = BA for all B we have

AB_1 = B_1A \Rightarrow \begin{bmatrix} a&0 \\ c&0 \end{bmatrix} = \begin{bmatrix} a&b \\ 0&0 \end{bmatrix} \Rightarrow b = c = 0

Similarly we have

AB_2 = \begin{bmatrix} a&b \\ c&d \end{bmatrix} \begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix} = \begin{bmatrix} 0&a \\ 0&c \end{bmatrix}

and

B_2A = \begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix} \begin{bmatrix} a&b \\ c&d \end{bmatrix} = \begin{bmatrix} c&d \\ 0&0 \end{bmatrix}

Since AB = BA for all B we have

AB_2 = B_2A \Rightarrow \begin{bmatrix} 0&a \\ 0&c \end{bmatrix} = \begin{bmatrix} c&d \\ 0&0 \end{bmatrix} \Rightarrow a = d

The result is that we have

A = \begin{bmatrix} a&0 \\ 0&a \end{bmatrix} = a \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix} = aI

for some a (which could be zero); in other words, A is a multiple of the identity matrix I.

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.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s