Matrices are math objects where the numbers are organized into a nice rectangular array of rows and columns. They are useful to learn because we can turn a system of equations into matrix form. We can manipulate matrices more easily than a collection of equations. To get you on the right track, we will look at matrices in this video lesson, their proper notation, their allowed math operations, and how to tell when two matrices are equal to each other.
To begin, we look at the proper notation for matrices. Look at this properly written matrix:
Proper notation for matrices
Our numbers are arranged in neat rows and columns. Surrounding our matrix are square brackets. These square brackets tell us that this particular group of numbers belongs together in one matrix. We can also label our matrix by calling it matrix A. If we have labeled our matrix, we can either write out the matrix with all the numbers in it or we can write it using our label surrounded by square brackets. Our matrices are also described by their size. This particular matrix has three rows and three columns, so we can also call it a 3x3 matrix. A matrix with four rows and two columns is a 4x2 matrix.
Okay, now that we know what proper matrix notation looks like, let's talk about when two matrices are equal to each other. You know how when two numbers are the same, they are the identical number, like when you have 2 and 2. Well, matrices are similar, but there is just a bit more involved. When two matrices are the same, all the numbers must be the same in the same positions and the matrices must both be the same size. For example, these two matrices are the same:
They are both the same size and all the numbers are the same in the same spots. They are both 2x3 matrices.
Take a look at these matrices. Are these the same?
These matrices are not equal.
They are not equal matrices because they are different sizes. Just because two matrices have all the same numbers does not mean they are equal. Their sizes must also be the same.
Matrix Addition and Subtraction
Let's cover the kinds of math operations we can do. We can do addition and subtraction. Matrix addition and subtraction is the same as number addition and subtraction. We add or subtract number by number. Because we have to match our number to number, our two matrices must be the same size, like this:
Matrices must be the same size to add or subtract.
We matched the numbers together. We matched the number in the first row and first column in the first matrix to the number in the first row and first column in the second matrix and so on. We can't add or subtract two matrices that are different sizes. We can however add or subtract the same number to all the numbers in a matrix.
Adding the same number to all the matrix numbers
We perform subtraction in the same way as addition. The two matrices must be the same size and we subtract number to number, matching the location of the numbers together. We can also subtract the same number from all the numbers in a matrix.
The last math operation we can do is matrix multiplication. There is no such thing as matrix division. Matrix multiplication is more complicated than number multiplication. You can easily multiply 3 and 5 to make 15. But with matrices, when we multiply two matrices together, we have to use a combination of multiplication and addition. Also, the number of columns in the first matrix must match the number of rows in the second matrix. So, we can multiply a 1x3 matrix with a 3x2 matrix, but we can't multiply a 1x3 matrix with a 2x3 matrix. And unlike numbers where the order of multiplication doesn't matter, with matrices, which matrix comes first matters.
Let's look at how we multiply two matrices together:
Multiplication with matrices
To multiply these two matrices, what we do is we match rows in the first matrix to columns in the second matrix. We then take each pair of numbers one by one, multiplying as we go, and then adding all the products together in the end. So for our matrices, we take the first row in the first matrix and match it up with the first column in the second matrix. This will give us the number in the first row and first column in the answer matrix. We match the 1 to the 0 and the 2 with the 2. We multiply 1 with the 0, the 2 with the 2, then add it all up. We have 1(0) + 2(2) = 0 + 4 = 4. The number in the first row and first column in the answer matrix is 4.
We have two columns in the second matrix, so we need to match the first row to the second column as well. We need to match each row in the first matrix to each column in the second matrix. Matching the first row in the first column to the second column in the second matrix gives us the number in the first row and second column in the answer matrix. Matching the numbers again we have: 1(1) + 2(3) = 1 + 6 = 7.
Now that we've matched the first row in the first matrix to every column in the second matrix, we now move on to the second row in the first matrix. And we do the same, matching it to every column in the second matrix. The number in the second row and first column in the answer matrix is: 3(0) + 5(2) = 0 + 10 = 10. The number in the second row and second column in the answer matrix is: 3(1) + 5(3) = 3 + 15 = 18. Our final answer is this matrix:
Final multiplication answer
As you can see, matrix multiplication is just a bit different than number multiplication. Remember that you are matching the rows in the first matrix to columns in the second matrix. We multiply the pairs of numbers and then we add them all up.
We can also simply multiply all the numbers in a matrix by one number. In this case, every number in the matrix is multiplied by the same number.
Let's review what we've learned now. Matrices are math objects where the numbers are organized into a nice rectangular array of rows and columns. We label them with either a letter surrounded by square brackets or by their size, such as a 3x2 matrix, which tells us that there are three rows and two columns.
We can do addition, subtraction, and multiplication to matrices. For addition and subtraction, the matrices must be the same size, and we simply add and subtract the numbers that are in matching locations in the two matrices. For matrix multiplication, we match each row in the first matrix to each column in the second matrix. We multiply each pair of numbers and then add them all up.
Watch and rewatch this lesson on matrices to be certain you can:
- Provide the definition of matrices
- Identify proper matrix notation
- Determine when matrices are equal
- Add, subtract and multiply matrices