Hello. In the previous video, we learned about solution of simultaneous linear equations using three different methods. In this video and the next one, we introduce a fourth method which is using matrix form. So in this video, we're going to learn how to set up a matrix form from a set of linear equations, and in the next one, we learn how to generate the solution.

The matrix algebra is very convenient for solution of simultaneous systems, and it doesn't require a lot of steps compared to the previous methods. In fact, it's very popular procedure for other situations even like interpolation, integration, and differential equations. We can simplify the problem to reduce to a set of simultaneous linear equations, and from there we solve a simple matrix problem to get the solution.

So let's start by an intuitive example to learn how to set up a set of equations into a matrix format. Let's start with this example where we have three equations in three unknowns: X, Y, and Z. Again, those unknowns in our circuit can be voltage and current at different nodes and branches. Okay, I can put a matrix for that equation. On the right hand side, I have a matrix that has three rows - one for each equation that we have - and then a vector on the right hand side that has the results for each of those equations.

Now you can see that the left hand side matrix is actually a linear combination of the three variables. So what we can do here is we split the equation into three equations added together. So I have a column associated with the variable X, and first column is the coefficient from the first equation. Second row has the coefficient from the second equation, and third row is the coefficient from the third equation. I do the same for Y and Z. So here I have the variable X times those coefficients associated with it, same for Y and Z, and on the right hand side I have the results.

Okay, now once I have this format, I can write it into a vector matrix multiplication. So I have a vector that has my unknowns and then a matrix that has the coefficients. So this would be the coefficient of X from the first equation, this is the coefficient of Y from the first equation, and this is the coefficient of Z from the first equation. So that when I do a matrix multiplication like that, it would be 25X + 5Y + Z equals to 106.8. Same for the second and third row, we can generate the second and third equations that we originally started with.

So in the previous example, we have seen how to go from a set of equations to a matrix format. So basically we're going to have a matrix that contains the coefficients from our equations, and then we have a vector that has our unknowns on the left hand side, and then the right hand side would have the results for each equation (right hand side value).

Next, let's see the generalized form on how to go from a set of linear equations to a matrix representation. So in general, if I have M linear equations with N parameters (so it's equation number one, equation number two, ..., equation number M), I can convert this set of equations into matrix multiplication as well.

First of all, I need to have a vector that has all of my parameters or unknowns. So here our parameters are X1, X2 through XN. In our circuit class, those would be voltages and currents that we'd like to solve for. And then we're going to have a matrix. This matrix would have a size of M by N - M is the number of equations and N is the number of parameters or unknowns. So we have M rows where each row corresponds to one equation. The columns of each row will contain the coefficients of the corresponding equation. So the first row will have coefficients coming from the first equation.

Then on the right hand side, we're going to have an M by 1 vector that contains the result, and the result would be simply the values that we have on the right hand side of each equation (C1, C2, ..., CM). Remember we have one equation corresponds to one row in that matrix. Okay, so this is in general our matrix format.

In our presentation, I can give variable names. So this matrix here we can call it matrix A (our coefficient matrix). This vector here I can call it X (this is the vector of our unknowns), and this is vector C which is the vector that we have for our results. And as a linear system of equation in a matrix representation, this can be written as AX = C.

So this matrix A is called the coefficient matrix. This will have all coefficients that we have from the equation. C is the right hand side vector that has the result, and X is the unknown vector that we'd like to solve for.

Another representation is not that different - it's called the augmented format. Maybe you have used this when you used iterative solution. With the augmented format, we're going to have this one big matrix that contains the matrix A and then we add the column C at the end here.

So for example, this is our matrix representation of the previous problem. The augmented format would be simply you copy A and then you add this extra column that corresponds to C. And then if you use the iterative method, this might be helpful to get the solution. Okay, but for our course, we're going to stick to the standard method AX = C, and we're going to learn next video how to solve for the vector of unknown X.

Okay, so far we have talked about the representation. So we put our three equations into a matrix format. So this matrix here was obtained from three equations. So equation number one is 25 (so I have the coefficient of X) 25X + 5Y + 1Z equals to 106.8. I can do the same with the second row - this gives me the second equation: 64X + 8Y + Z = 177.2. Third row gives me the third equation: 144X + 12Y + Z = 279.2.

So, we started from the equations. But It's preferred for us to go to the matrix representation because it is easier to solve for X. So far, we learned how to get this matrix representation, but we don't know how to solve for the vector of unknowns [X Y Z]'. To do so, we're going to use a matrix inverse, and this should be the topic of our next video.

So to conclude this video: we started by a linear set of equations, we learned how to obtain a matrix representation AX = C. Simply, this matrix of coefficients contains all the coefficients that we have in our equation. This is a vector of our parameters, and this is the right-hand side of the equation. In the next video, we're going to learn how to use matrix inverse to solve for the vector of unknown X.

Thank you.