That Makes It Invertible!

There are several ways of determining whether an $n \times n$ matrix ${\bf A}$ has an inverse:

$\det {\bf A} \ne 0$
The span of the row vectors is $\mathbb{R}^n$
Every matrix equation ${\bf Ax} = {\bf b}$ has a unique solution
The row vectors are linearly independent
When applying Gaussian elimination, ${\bf A}$ reduces to the identity matrix ${\bf I}$
The only solution of ${\bf Ax} = {\bf 0}$ is the trivial solution ${\bf x} = {\bf 0}$
${\bf A}$ has only nonzero eigenvalues
The rank of ${\bf A}$ is equal to $n$

Of course, it’s far more fun to remember these facts in verse (pun intended). From the YouTube description, here’s a Linear Algebra parody of One Direction’s “What Makes You Beautiful”. Performed 3/8/13 in the final lecture of Math 40: Linear Algebra at Harvey Mudd College, by “The Three Directions.”

While I’m on the topic, here’s a brilliant One Direction mashup featuring the cast of Downton Abbey. Two giants of British entertainment have finally joined forces.

Predictions

Dear writers of elementary math problems: If you ask children to “predict how many heads and tails you will get if you flip a coin 10 times,” there is no single correct answer since even a prediction of 10 consecutive heads is valid (if not the best). Please use some other verb besides “predict.” Thank you.

25 divided by 5 is 14

Teaching base 10

Price (on the day of this writing) on Amazon.com for a base-10 starter kit, featuring 100 small plastic blocks, 30 rods (representing 10), 10 plates (for 100), and one big cube (for 1000): $31.32.

Cash value of 100 pennies, 30 dimes, 10 dollar coins, and a $10 bill: $24.00.

Analog clocks

What time is it?

From the perspective of an elementary student — even a good student who generally can read the hands of a clock correctly — the answer probably is 2:50. After all, the hour hand is pointing much closer to the 2 than to the 1.

To address this misconception, perhaps the best suggestion that I’ve heard is using the analogy of waiting for your birthday. A child doesn’t turn 7 until his (or her) birthday comes. On the day before his birthday, he still has to say that he’s 6. Likewise, if the minute hand is just a few minutes before 7:00 and the hour hand is not all the way at seven, we still have to say that the time is six-something.