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. So here I have 3 I just be adding eight, here I have -2 adding -2 becomes -4, and lastly 13+-2 is 11. 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. 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. We can do the same thing with Matrices as well. What we could do is we could say okay let's just add these two equations together and see what we come up with.
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. 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. 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. That's still the same equation as long as we distribute that 2 all the way through. 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. 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. It doesn't matter which one I put first it's the same system. So the first row operation we can do is switch the order of the rows thinking about our equations right here. And those are called row operations, how we can manipulate these rows okay. 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. Okay so taking this information and just translating into a matrix form. And typically a dotted or shadow line is drawn down the middle to separate our variables from our answers okay.Īnd 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. So we have our 3 and our x and -2, and then our 13 same thing for the second equation 4, -1, -1. 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. Okay so the first thing is taking this information and putting it into matrix form okay. But we're going to talk about now is how we can do this with a matrix. So behind me I have a system of linear equations, okay we know we can solve this using elimination or substitution.
Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank hence in such a case there are an infinitude of solutions.Īn augmented matrix may also be used to find the inverse of a matrix by combining it with the identity matrix.Solving a system of linear equations using an Augment in matrix. The solution is unique if and only if the rank equals the number of variables. Specifically, according to the Rouché–Capelli theorem, any system of linear equations is inconsistent (has no solutions) if the rank of the augmented matrix is greater than the rank of the coefficient matrix if, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. This is useful when solving systems of linear equations.įor a given number of unknowns, the number of solutions to a system of linear equations depends only on the rank of the matrix representing the system and the rank of the corresponding augmented matrix. The augmented matrix ( A| B) is written as In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.