Consider the circuit shown in the figure. The currents I1, I2, and I3, in amperes, are the solut

step 1 In this problem, we are given the system of equations 2. I to the 1 plus 4, I to the 3 is equal

Consider the circuit shown in the figure. The currents I1,...

Key Concepts

Kirchhoff's Circuit Laws

Kirchhoff's circuit laws, which include the current law (KCL) and the voltage law (KVL), are fundamental principles in electrical engineering that govern the conservation of charge and energy in electrical circuits. These laws lead to the formulation of systems of linear equations that model the behavior of electrical networks, making them essential for analyzing and solving practical circuit problems.

System of Linear Equations

A system of linear equations consists of multiple linear equations that share common variables, where the solution represents the set of variable values that satisfy all equations simultaneously. In many engineering and scientific problems, including those in circuit analysis, systems of equations arise naturally when applying conservation laws or other principles, and solving these systems is key to understanding the behavior of the system as a whole.

Matrix Inversion Method

The matrix inversion method involves representing a system of linear equations in matrix form and then solving for the unknowns by multiplying both sides by the inverse of the coefficient matrix, provided that the matrix is invertible. This technique is particularly useful for efficiently solving complex systems, especially when the system size is moderate and when computational tools for matrix operations are available.

Coefficient Matrix

In the context of a system of linear equations, the coefficient matrix contains the coefficients of the variables from each equation arranged in a structured grid. This matrix plays a central role in linear algebra techniques, including the matrix inversion method, as it encapsulates the contributions of each variable and allows for a compact representation of the system.

Transcript

00:01 In this problem, we are given the system of equations 2.

00:05 I to the 1 plus 4, i to the 3 is equal to e1.

00:11 And then i to the 2 plus 4, i to the 3 is equal to e2.

00:17 And finally, i to the 1 plus i to the 2 minus i to 3 is equal to 0.

00:22 So it says to solve this using inverse of matrices.

00:25 So the first thing i need to do is a coefficient matrix.

00:29 Two, there's not an i sub two term, so i wrote zero, and then i do the four.

00:36 In the second equation, there's not an i sub one term, so that's going to be a zero coefficient.

00:41 There's a one in front of this one, and then the four, and finally a one, a one, and a negative one.

00:48 That's my coefficient matrix.

00:49 I need to multiply that by my variables, which are i sub one, i sub two, i sub three, and that's going to equal my answer matrix.

00:58 It says e1 is equal to two.

01:00 28, e2 is equal to 21.

01:02 It said that in the problem.

01:03 And then my final answer is zero.

01:06 This is way easier to do on the calculator.

01:11 You're simply going to multiply the inverse of the coefficient matrix because we can't divide by matrices times the answer matrix.

01:22 Finding the inverse of a three by three involves many steps.

01:25 So by definition of inverse, on the right, i'm sorry, on the left side, we're going to get the identity matrix.

01:34 Because when you multiply a matrix times it's inverse, you get the identity matrix.

01:39 And we still have our variables right here.

01:42 Over here, i went ahead and wrote it in the correct order so you can actually multiply, and we've got to find the inverse of our coefficient matrix right here.

01:52 Once we find that, it's easy to multiply by the answer matrix.

01:58 So i need to find what i call a new matrix to help find the inverse, and that is going to be expansion by my answer matrix.

02:05 Miners.

02:07 So i need to for the first element cross out this row and this column and i end up with one for one negative one.

02:16 I need to find that determinant in just a second.

02:19 When i cross out the second column and the first row for element zero, i get zero for one negative one.

02:27 I've gone through and done that for the entire matrix.

02:31 So this this third term right here, i'm going to cross out this column and this row and i get 0 -1 -1 -1 and it follows when i cross out the first row and the second column i get 0 -4 -1 -1 when i cross out my middles i get 241 -1 -1 -1 and when i cross out the last column and the middle row i end up with the 2 .11.

03:06 Finally, when i cross out this first column and this bottom row, i get 0414.

03:15 When i cross out the middle column and the bottom row, i get 2404.

03:22 And finally, when i cross out this last row in this last column, i get 2 .01.

03:28 All of these are determinants.

03:30 One times negative 1 minus 4.

03:33 That's going to be negative 5.

03:34 0 minus 4 is negative 4.

03:37 0 minus 1 is negative 1.

03:39 0 minus 4 is negative 4.

03:42 Negative 2 minus 4 is negative 6.

03:45 2 minus 0 is 2.

03:51 8 minus 0 is 8.