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Augmented Matrices - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Some of the most common algebra mistakes come from misunderstanding the order of operations or what is commonly known as PEMDAS. We must keep in mind that the multiplication and division are done at the same time in the order of operations, as are addition and subtraction. The most important way to help not make these common algebra mistakes is by practicing and being aware of pitfalls.

Solving a system of linear equations using an Augment in matrix. So behind me I have a system of linear equations, okay we know we can solve this using elimination or substitution. But we're going to talk about now is how we can do this with a matrix.
Okay so the first thing is taking this information and putting it into matrix form okay. And how we do that is basically by taking each element of our linear equation its coefficient and throwing it into its own space in a matrix. So we have our 3 and our x and -2, and then our 13 same thing for the second equation 4, -1, -1. And typically a dotted or shadow line is drawn down the middle to separate our variables from our answers okay.
And really what this is representing is this first column is our x values, our second is our y's and our third is our answer. Okay so taking this information and just translating into a matrix form. Now what we have to remember is we are, this is just a system of equation so anything we can do do in an equation we can also do to this matrix. And those are called row operations, how we can manipulate these rows okay. So the first row operation we can do is switch the order of the rows thinking about our equations right here. It doesn't matter which one I put first it's the same system.
Okay so one row operation we could do is to switch our rows, the second equation becomes first and the first equation goes to the second and again the dotted line to switch it. Okay going back to our equations we could just as easily say that this equation we can multiply by two right so this way we could say let's multiply this by two and then this becomes 6x-4y is equal to 26. That's still the same equation as long as we distribute that 2 all the way through.
We can do the same thing with Matrices and that's called a scale of multiple okay you can use the scale of multiple basically I could say okay let's multiply this by two that top equation then becomes a -2, -2, the bottom equations stays the same 3, -2, 13 okay.
The last row operation we can do is adding and subtracting rows together, okay we did this in elimination we go back over here let's ignore that 2 that we multiplied by we can try to get rid of a variable. Okay typically we would try to line it up so our x's and y's canceled but that was just a special case in solving it. What we could do is we could say okay let's just add these two equations together and see what we come up with. We can do the same thing with Matrices as well. Okay and typically how I at least designate that and how your teacher does it might be different is just to say okay next to row 2 just say like okay let's add row one.
Okay and so what that would tell me is that my lets go over here first row says the same 8, -2, -2, and then my bottom row I'm adding the corresponding component from the first row. So here I have 3 I just be adding eight, here I have -2 adding -2 becomes -4, and lastly 13+-2 is 11. So this tells me that I am just adding these two rows together, I could just as easily said okay let's ignore this and I could say okay let's subtract 2 times row one. And that would tell me that I would need to multiply row one by 2 and subtract it, cause this is just a again a scale of multiple of something you can always multiply by a constant. So and recap, we took our equation system of equations and turned it into a matrix and then from there there's a number of matrices number of matrix operations that we can use are called row operations we can switch the order of our rows, we can multiply by a scalar or we can add rows together or add multiples of rows as well.

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