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Amy has a master's degree in secondary education and has taught math at a public charter high school.
Matrices are everywhere in math. What is a matrix? It is an array of numbers arranged in rows and columns. Just think of numbers arranged nicely in a rectangular grid. Matrices come in all sizes. We call a matrix with 3 rows and 4 columns a 3 x 4 matrix. Yes, a matrix is defined by its size, the number of rows and columns, and when we give the size of the matrix, the rows always come first.
You will see these more and more often as you progress in your math career. Being able to perform operations on them will be very helpful to you. There are only three row operations: switching, multiplication, and adding. It is these three row operations that we will be looking at in this video lesson. I encourage you to come up with your own matrix and try out these three operations as we are talking about them.
The first row operation is switching. This operation is when you switch or swap the location of two rows. In this matrix, we can switch the first and third rows so that the 1 moves to the top.
The goal of switching is to get a better organized matrix. What I've done here is move the row with the leading 1 to the first row so that we have our first row beginning with a 1 and zeroes underneath the 1. Our matrix now begins with 1 in the first row, 0 in the second row, and 0 in the third row.
We can use arrows between our original matrix and the new matrix to show how we have switched the rows. For our matrix, we can put an arrow from the third row in the original matrix to the first row in the new matrix to show that we have switched the first row with the third.
Our next operation is multiplication. When we use this operation, we multiply one row with a certain number. The goal of multiplication is to get a better organized matrix. When we multiply one row by a certain number, we make sure we multiply each digit of our row with that number. For example, in this matrix, we can multiply the second equation by 1/5 to change the first non-zero number to a 1.
To show what you've done, you can write a little bit of notation between the beginning matrix and the end matrix. For what we have done, we can draw an arrow from the second row to the new second row and write (1/5)R sub 2 to show that we have multiplied the second row by 1/5.
The third and last operation is addition. This is when we add two rows together. The goal of adding two rows together is to simplify our matrix. We use addition when we can change a particular number to 0. For example, in this matrix, we can add the third row to the second row to change the first number in the second row to a 0.
To help us keep track of all of our changes, we note this operation by drawing an arrow from the beginning second row to the new second row and write on top of the arrow R sub 2 + R sub 3 so we know that we added row two to row three.
One thing to note here is that when we add two rows, we can choose which row to replace. If we are adding rows two and three, we can either replace row two or row three with the new row. We choose the one that will give us the simpler matrix.
We've covered all three operations separately. Let's combine them now. Yes, we can combine our operations to help us better organize and simplify our matrices. The most common combination is multiplying a certain row by a number and then adding it to another row to form a new row. For example, in this matrix, we can first multiply our first row by -2 and then add it to the third row so that the first number in the third row becomes 0.
To note this operation, we can draw an arrow from the beginning third row to the new third row and write on top of it -2R sub 1 + R sub 3 to tell us that we multiplied the first row by -2 and then added it to the third row. Notice because we are combining our operations, we left the first row the same even though we multiplied it by -2. This is perfectly okay.
Let's review what we've learned now. We've learned that a matrix is an array of numbers arranged in rows and columns. A 3 x 2 matrix will have three rows and two columns. There are only three row operations that matrices have. The first is switching, which is swapping two rows. The second is multiplication, which is multiplying one row by a number. The third is addition, which is adding two rows together.
For each operation you do, it is recommended that you make note of what you did in between your matrices by drawing an arrow and writing little notes on top of the arrow. The goal of all these combined operations is to better organize and simplify your matrix.
You can also combine the operations to help you simplify your matrix. The most common combination is to multiply one row by a number and then add it to a different row.
Once you are done with this lesson you should be able to:
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Back To CourseAlgebra II Textbook
26 chapters | 256 lessons