# High School: Number and Quantity

### Vector and Matrix Quantities HSN-VM.C.10

10. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

Students should know what the zero and identity matrices are and how they're used in matrix addition and multiplication.

First, we'll start with zero. It's a great number, right? You can add zero to anything, and you'll just get the same number back. For instance, 6 + 0 = 6 and 0 + 19 = 19. It's called the additive identity because we can add zero to any number without changing that number's "identity." (So when it goes to see a move that's rated R, adding zero won't help it if it's under 17.)

In a similar fashion, 1 is the multiplicative identity. We can multiply any number by 1 and not change the "identity" of the number. For example, 3444 × 1 = 3444 and 1 × 7 = 7.

Yeah, but that's the realm of numbers. We're talking about matrices, here.

Students should know that the zero matrix is exactly what they think it is: a matrix filled with zeros. When added to any other matrix, it yields the other matrix as the sum.

Multiplication, on the other hand, is a little trickier.

The identity matrix is one that, multiplied by matrix A, yields matrix A. Because of the way matrix multiplication works, the identity matrix is not full of ones; instead, it has a diagonal of ones, starting at the top left hand corner and going down. All the other entries are zeros.

So one example of an identity matrix looks like this

And another one is

You get the picture.

If a square matrix is multiplied by its multiplicative inverse, it yields that identity matrix. (We want the same number as rows as columns, because we want it to be commutative—it doesn't matter whether we put matrix A or its inverse first, we want to be able to multiply them to get that identity matrix.)

We can find the multiplicative inverse of any matrix pretty simply. If we have a matrix , then the multiplicative inverse is

A matrix of  becomes .

Students should also know what a determinant is and how to calculate it.

To find the determinant of a 2 × 2 matrix, simply multiply the two diagonals. Then subtract the one that starts top left minus the one that starts top right. The result is our determinant.

For example, given the matrix , we can calculate the determinant this way: 7 × 2 – 3 × 8 = -10. That's really all it takes.

Finding the determinant of a 3 × 3 matrix is a bit more complicated. Let's say we need to find the determinant of

Here's what we do.

First, recopy the first two columns and attach them onto the end. We should have 5 columns and 3 rows. In other words, we should have something that looks like this:

Then, working from the top left toward the bottom right, multiply the entries in each of the 3 diagonals, and add those 3 numbers. From the top left, we get:

1 × 0 × 0 + 2 × 1 × 5 + 3 × 4 × 2 = 34

Now work from top right towards bottom left, and do the same thing. From the top right, we have:

2 × 4 × 0 + 1 × 1 × 2 + 3 × 0 × 5 = 2

So far, so good. Then, Subtract the second number from the first number. That means 34 – 2 = 32. That is our determinant.

#### Drills

1. Why is zero the additive identity element?

Because zero added to any number equals that same number

The identity element is a number that doesn't change the identity of another number when a given operation is performed. So an additive identity element is a number that doesn't change the identity of another number when it's added to it. It's zero because when you add zero to a number, the result is that same number.

2. What is the multiplicative inverse of the following matrix?

Remember, the formula is . If we substitute everything in properly, we end up with . If we distribute the scalar as needed, we'll end up with (C) as our answer.

3. What does a 5 × 5 identity matrix look like?

If a matrix A is multiplied by the identity matrix, the result is matrix A. We could either create a matrix A and fill it with numbers, or we could use the knowledge that all identity matrices have ones not all throughout, but in a diagonal from the top left to the bottom right. That means (B) is our 5 × 5 identity matrix. Give it a high-five (by five)!

4. What is the multiplicative inverse of the following matrix?

The multiplicative inverse of a matrix is . The value of  becomes , and multiplied into the matrix (not The Matrix, of course), we get (A) as our answer.

5. What is the determinant of any identity matrix?

1

A 2 × 2 identity matrix has a determinant of 1, as does a 3 × 3 identity matrix. If we keep going, we'll never end. Since identity matrices have one single diagonal of ones from the top left to the bottom right with all the rest zeros, we'll have a lot of zeros to add and subtract when we find the determinant. But zeros don't change the identity of the number when we add them, so it'll always be 1 (and as we all know, one is the loneliest number).

6. How is it possible for a matrix not to have a multiplicative inverse?

The matrix must be filled with zeros

A multiplicative inverse of a matrix is one that gives the identity matrix when multiplied with the original matrix. If we find the multiplicative inverse of an identity matrix, we get the identity matrix. So it does have a multiplicative inverse; it's just itself. But if we find the multiplicative inverse of a matrix full of zeros, we get a matrix full of zeros. A bunch of zeros and a bunch of zeros result in—you guessed it: a bunch of zeros (as in, not the identity matrix).

7. What is the determinant of the following matrix?

15

If we multiply the diagonals, well get 2 × 3 = 6 and -1 × 9 = -9. Subtracting them in the proper order, we get 6 – (-9) = 15. That's our determinant.

8. What is the determinant of the following matrix?

20

If we multiply the diagonals and subtract, we'll have 5 × 4 – 0 × 1 = 20 – 0 = 20. Simple enough, right?

9. What is the determinant of the following matrix?

-5

The determinant of a 3 × 3 matrix is a bit more complicated, but we're sure you can handle it (probably because you already have). If we follow the rules, we should end up with the top left diagonals giving us 2 + 12 + 0 = 14 and the top right ones equaling 0 + 12 + 7 = 19. Subtracting the two gives us -5.

10. What is the determinant of the following matrix?