Topic 4.11: The Inverse and Determinant of a Matrix

Two instruments, two channels

This band is just two people, one on drums and one on bass, and everything you’re hearing right now comes out of those two instruments. Out here in the middle of the crowd, the sound engineer is recording the whole show, but only as a stereo file, which means just two channels, left and right, and both instruments are blended into both of them.

So, once they’re blended together like this, can you ever get the bass back out on its own? With the right four numbers, you can. And it isn’t a rough guess, either, because that’s the real bass line, every single note of it, pulled straight back out of the mix. But earlier tonight, at soundcheck, the engineer tried one mix that nobody can undo exactly, not with any four numbers, and not on any computer in the world.

By the end of this lesson, you’ll be able to work out these four numbers yourself and pull this band back apart into its two instruments. And you’ll be able to look at any mix and tell, in about two seconds, whether it can be undone at all. So let’s go back a few hours to soundcheck, while the sun’s still going down, and start with the easiest mix there is — the one that doesn’t mix anything at all.

The mix that mixes nothing

Both instruments are plugged straight into the desk, and the desk saves its settings, so we’ll always know exactly how a mix was made. A real song with twenty tracks and echoes can’t be pulled apart like this, and by the end of the lesson you’ll see exactly why. These four knobs make what’s called a matrix mixer, and yes, that’s really what sound engineers call it.

Each knob says how much of one instrument goes into one channel, so the top row is the left channel’s recipe, the bottom row is the right channel’s, and each column belongs to one instrument. Right now the left channel gets two parts drums and one part bass, and the right channel gets one part of each, so both speakers play a blend.

Can you set these four knobs so the left speaker plays only the drums and the right speaker plays only the bass, each one exactly as loud as the real thing? Let’s write those four numbers down exactly the way they sit on the desk.

This is called the identity matrix, and we write it I. It has ones on the diagonal that runs from the top left to the bottom right, and zeros everywhere else. Read across the rows, and the left channel is one times the drums plus zero times the bass, while the right is zero times the drums plus one times the bass, so nothing gets mixed at all.

So what does I do to another matrix? Let’s multiply two, five, one, three by the identity, row times column, the way we’ve multiplied matrices before. The top left is two times one plus five times zero, which is two, and every other entry works the same way, because the ones pick out each number and the zeros wipe out the rest. So we’re right back where we started.

It works in the other order too, so I times this matrix gives the same matrix back. Multiplying by the identity changes nothing, the same way multiplying a number by one changes nothing.

Undo the mix

Now let’s put the engineer’s first real setting back — two parts drums and one part bass on the left, and one part of each on the right — and let’s call this mix A. Let’s freeze the song at one single instant, where the drum track’s value is two and the bass track’s value is one.

Then the left channel holds two times two plus one, which is five, and the right channel holds two plus one, which is three. So if all you have is the five and the three, can you work your way back to the two and the one?

The recording says that the left channel is two drums plus one bass, and the right channel is just drums plus bass. I want to get rid of the bass, and both channels have exactly one part of it, so let’s subtract, left minus right, and the bass cancels out and leaves just the drums.

That’s actually an old karaoke trick, because subtracting one channel from the other cancels anything that sits equally in both, like a singer standing right in the middle. Then the bass is whatever’s left of the right channel once the drums are gone, R minus the drums, and since the drums are L minus R, that’s R minus L plus R, or two R minus L.

Now collect those four numbers into a matrix, with one and negative one on the top row and negative one and two on the bottom. That’s the unmix, and we call it A inverse. Let’s test it on our instant. Five minus three is two, and two times three minus five is one, so we’re back to drums two and bass one.

And it works for every instant, not just one. Put those four numbers into the unmix patch and let the song play. Out of the two channels come the drums alone and the bass alone, landing exactly on top of the originals.

So what does the unmix do to the mix itself? Let’s multiply A inverse by A. Row times column, two minus one is one and one minus one is zero, then negative two plus two is zero and negative one plus two is one.

Mix, then unmix, and you get the identity, the mix that does nothing, which is exactly what undoing something means. And A times A inverse is the identity too, so whichever way round you multiply them, you get I.

Flipping each number upside down doesn’t work, though. One half, one, one, one might look like an inverse, but multiply it by A and you get nowhere near the identity. Here’s a different mix, C, and someone hands you a matrix B and says it’s the unmix. Is B really the inverse of C?

The mix nobody can undo

Then, as it got darker, the engineer tried putting both instruments in the same spot at the same level, one part into the left channel and two parts into the right. Look at the screen behind the band, which draws every instant of the sound as a dot of light. A moment ago the song made a cloud, and now it’s collapsed onto one single line.

Let’s freeze it at drums two and bass one again. The left channel gets two plus one, which is three, and the right channel gets twice as much, six. Now make it drums one and bass two instead. The left channel is one plus two, still three, and the right channel is still six. Two completely different moments, and they make exactly the same recording.

Can you find another drums and bass moment that makes exactly this same recording, three and six? In fact there’s a whole line of them, and every single one gives three and six, so if all you have is the recording, nobody can know which one the band actually played.

A second screen by the desk shows why. Each instrument’s column becomes a vector, an arrow with how much goes left measured across and how much goes right measured up, and here both columns are one, two, so both arrows point the same way. Drums two and bass one means two drum arrows plus one bass arrow, tip to tail, and since every one of them goes one across and two up, they land on three, six. And so does one drum arrow plus two bass arrows, because they’re really the same arrow.

Remember inverse functions? A function can only be undone if no two inputs share an output, and here a whole line of different inputs all share one output, so no unmix can possibly exist. Read the rows, too. The right channel’s recipe, two and two, is exactly double the left channel’s, so the right channel is just a louder copy of the left, and that’s two files but only one recording.

It still sounds like music, and you can still multiply by this matrix as much as you like. It just can’t be undone. And that’s the trouble with a real song, too, because when twenty tracks get squeezed into two channels, there are always lots of different performances that make exactly the same recording.

How much room between the arrows

So how can you tell from the four numbers alone whether a mix can be undone? Here’s the next try, with the drums three parts left and one part right, and the bass one part left and two parts right.

Fill in the parallelogram between the two arrows like a sheet of light, and you can think of its area as how much room the mix leaves between the two instruments. This one has an area of five, against one for the identity’s little square, and the mix we couldn’t undo had no room at all, because its sheet was flat.

Let’s check that five ourselves. Put a box around the parallelogram, four wide and three tall, so the box has an area of twelve. Then cut away the six pieces around it, two triangles of one and a half each and four more pieces of one each, which add up to seven, and twelve minus seven is five.

There’s a much faster way. Multiply down the main diagonal, three times two is six, then subtract the other diagonal, one times one, and six minus one is five again. That number is called the determinant. For any two by two matrix a, b, c, d, the determinant is a d minus b c, and we write it det of A. And its size, ignoring any minus sign, is always the area of the parallelogram that the two columns make.

Try it on the mixes we’ve met. The identity gives one times one minus zero, which is one, and our first mix gives two minus one, also one. And the mix nobody could undo gives two minus two, which is zero, because its two arrows were parallel and its sheet was flat.

Now it’s your turn to break a mix. Can you drag the bass’s arrow to a spot where this mix can’t be undone anymore?

Let’s put the bass back, and this time swap the two instruments’ places, so the drums go mostly to the right channel and the bass mostly to the left. As they pass, the sheet turns over, and the determinant comes out as negative five. But the instruments are just as far apart as before, so this mix can be undone perfectly well. The minus sign only says which instrument is on which side, and the area is still five.

Rows work the same way. Take the matrix two, one, three, negative one. Its columns make one parallelogram and its rows make a different one, but both have an area of five, because the determinant is negative five.

And that’s no accident, because using the rows instead of the columns just trades the b and the c, and a d minus c b is exactly the same number as a d minus b c. So a matrix has an inverse exactly when its determinant isn’t zero, and that one sentence is the whole story. Which one of these four mixes can’t be undone?

Why the inverse divides

We found the first unmix by thinking it through. But there’s a pattern that works for every two by two matrix, and it shows exactly where the determinant comes in. Take any matrix a, b, c, d. Swap the a and the d, and put minus signs on the b and the c.

Now multiply them. The top left comes out as a d minus b c, and the top right is minus a b plus b a, which cancels to zero. The bottom left, c d minus d c, cancels to zero the same way, and the bottom right, minus c b plus d a, is a d minus b c again. So we get the determinant on the diagonal and zeros everywhere else, which is the determinant times the identity. That means the swapped matrix is just the inverse multiplied by the determinant.

So divide by the determinant, and that gives the formula, A inverse equals one over a d minus b c, times d, negative b, negative c, a. And now you can see why a determinant of zero kills it, because you’d have to divide by zero. Try it on our first mix. Swap the two and the one on the diagonal, flip the signs of the other two ones, and divide by its determinant, which is one, and you get exactly the unmix we found by hand.

What happens if you forget to divide? Here’s the mix with determinant five, and a patch with just the swapped and negated numbers. The drums and the bass come out cleanly separated, but five times too loud, and the meters slam into the red. Divide by five, and everything settles exactly onto the originals.

So the inverse of this mix is one fifth of two, negative one, negative one, three, and since the swapped matrix gives five times the identity, one fifth of it gives exactly the identity. You don’t always have to do this by hand, because a graphing calculator or Desmos will find the inverse and the determinant for you, and you can multiply back to check.

Now you try one by hand, with this matrix Q, whose determinant isn’t five — and I’ll leave the formula up here so you can use it. What’s the inverse of Q?

Unmix the headliner

And now it’s night, the show’s on, and here’s the headliner’s mix, four parts drums and one part bass on the left, and two parts drums and three parts bass on the right. This is the recording from the very start of the lesson, and now you know everything you need to pull it apart.

Can you build the patch that pulls the drums and the bass back apart? And there’s the bass on its own, and the drums on their own, pulled out of one stereo recording by the four numbers of its inverse.

And here’s one last mix, two, six, one, three, and this one should take you about two seconds. Two times three minus six times one is zero, so you already know it can’t be undone, without trying a single number. And you’ve been hearing this for years, because since 1961, stereo FM stations have sent the left plus the right and the left minus the right, and every stereo radio undoes that matrix to give you two speakers back.

So here’s the lesson in four lines. The identity has ones down the diagonal, and multiplying by it changes nothing. The inverse undoes a matrix, so multiplying them either way round gives the identity. The determinant, a d minus b c, is the area the columns or the rows span once you drop its sign, and zero means they’re parallel.

And a matrix has an inverse exactly when its determinant isn’t zero, and then it’s one over the determinant times d, negative b, negative c, a. So whenever the determinant isn’t zero, there’s always an unmix, and now you can build it yourself.