Solving a Matrix

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A matrix can be solved in several different forms, with the most common form being the rho echelon form. Use Gaussian elimination to get to rho echelon form with help from a math teacher in this free video on basic math lessons.

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Video Transcript

So how does one solve a matrix? Hi, I'm Jimmy Chang. I've been teaching college mathematics for nine years. And solving a matrix is a little bit on the lengthy side, it's a rather long process depending on what the numbers are though. By now I'm sure you know what a matrix consist of, but how does one go about solving it. Well, what we're going to do is talk about a couple of forms that you want to use to solve a matrix. In other words a couple of forms in which you want the matrix to be in, and then we'll discuss briefly the methods that we'll take to get there. But show you the benefits more importantly of those particular forms. Now one of the more common forms of solving a matrix, the form that you want the matrix to be in in order for you to figure out the answers is that of Row Echelon form. Here's an example as to what Row Echelon form looks like. And what you here's a structure as to how that works. You may notice here that there's a series of ones that are in a diagonal fashion. That's an example of Row Echelon form. Now also, beneath each one, these are called leading ones by the way. But underneath each leading one is zeros. The more zeros you have the better, but if you have an entire row of zeros not a bad thing, but necessarily a good thing either. Because a row full of entirely zeros is not going to help you in terms of what your final answers are going to be. Now, to get to Row Echelon form you need what's called Gaussian elimination. Now, the series of row operations that will get you to this particular form, but here's how this works. Once you have a matrix in Row Echelon form you can convert this to X plus three Y, plus five Z equals to nine. In other words, you'll be able to convert the matrix into three, into various equations. The second equation is going to be Y plus two Z is equal to negative one, and the bottom equation can be written as Z is equal to six. Now the reason Row Echelon form is beneficial is because you know what the bottom answer is. You can take the six, plug it back into the second equation and find out what Y is. And then once you know your Y, you can take both the Y and Z values and plug it back in to figure out your X value. Now the other form is what's known as the reduced Row Echelon form. It's a simpler form, but it takes more time to get there. The method to get to that is Gauss-Jordan elimination, which means you have to use more row operations to get to what you want. Take a look though, look how clean everything is. You have leading ones along the diagonal, with zeros above and below it. But here's the beauty of it, you know what the answers are for each particular variable. In this case, X is five, Y is negative two, and Z equals to zero. But once you master those methods, you'll be able to solve the matrix very easily. So I'm Jimmy Chang and that is how you solve a matrix.


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