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.