Question

Solve each system of equations by calculator using the unit matrix method. Applications. Applying Kirchhoff's law to a certain three-loop network gives $$\begin{array}{l} 283 I_{1}-274 I_{2}+163 I_{3}=352 \\ 428 I_{1}+163 I_{2}+373 I_{3}=169 \\ 338 I_{1}-112 I_{2}-227 I_{3}=825 \end{array}$$ Solve this set of equations for the three currents

Short answer

\( I_{1} \approx 3.478 A, I_{2} \approx -0.851 A, I_{3} \approx -1.108 A \)

Step-by-step solution

- Setting up the matrix equation

Represent the system of linear equations in matrix form, where the matrix of coefficients is multiplied by a column vector of the variables to equal a column vector of the constants. The system can be written as \[ A \vec{I} = \vec{B} \] where \[ A = \begin{pmatrix} 283 & -274 & 163 \ 428 & 163 & 373 \ 338 & -112 & -227 \end{pmatrix}, \] \[ \vec{I} = \begin{pmatrix} I_{1} \ I_{2} \ I_{3} \end{pmatrix}, \] and \[ \vec{B} = \begin{pmatrix} 352 \ 169 \ 825 \end{pmatrix}. \]

- Using the unit matrix method with a calculator

Enter matrix A and vector B into the calculator. Then, to find the vector of currents \( \vec{I} \), use the unit matrix method (also known as inverse matrix method), where you calculate the inverse of matrix A and multiply it by vector B: \[ \vec{I} = A^{-1} \vec{B}. \] Calculate the inverse of A, \( A^{-1} \), with the calculator, and then multiply it by \( \vec{B} \) to find the solution vector \( \vec{I} \). Note that this step requires a calculator that can handle matrix calculations.

- Interpreting the result

After performing the matrix multiplication on the calculator, you should have the values for \( I_{1} \), \( I_{2} \), and \( I_{3} \). These values are the currents in amperes for each of the three loops in the network as determined by Kirchhoff's law.

Key concepts

Kirchhoff's Law

Kirchhoff's Law is a fundamental principle in electrical circuit analysis. It comprises two rules, Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL). KCL states that the total current entering a junction must equal the total current leaving the junction. This is based on the conservation of charge. KVL states that the total sum of voltages around any closed loop in a circuit must equal zero, reflecting the conservation of energy.

When dealing with complex circuits, such as the three-loop network in our example, these laws become a set of simultaneous linear equations reflecting the current flow and potential differences. Each equation represents a current path or voltage drop, and solving them collectively provides the current or voltage in each component of the circuit. Using Kirchhoff's Law transforms a physical problem into a mathematical one, making it solvable using methods for systems of linear equations.

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. In the context of Kirchhoff's laws, these variables can represent currents or voltages in an electrical circuit. To solve for these variables, it's often convenient to use matrix representation, where each equation becomes a row in a coefficient matrix, and the variables and constants form vectors.

By converting the linear system to a matrix equation, we simplify the problem and can use computational tools to solve it. The challenge now is not in the physical interpretation of the equations, but in the numerical methods we choose to use, such as the unit matrix method highlighted in our exercise.

Inverse Matrix

The inverse of a matrix, denoted as A^{-1}, is a unique matrix such that when multiplied by the original matrix A, it yields the identity or unit matrix, denoted I (where I is a square matrix with 1s on the diagonal and 0s elsewhere). The identity matrix acts much like the number 1 in multiplication, because when any matrix is multiplied by I, it remains unchanged.

However, not all matrices have an inverse. A matrix must be square (same number of rows and columns) and have a non-zero determinant to have an inverse. In the context of our electrical circuit problem, we can solve for the currents by finding the inverse of the coefficient matrix A and multiplying it by the constants' vector B. If A is invertible, this gives us a precise solution to the system of linear equations.

Matrix Multiplication

Matrix multiplication is a mathematical operation that produces a new matrix from two matrices, reflecting a linear transformation. One important thing to remember is that matrix multiplication is not commutative, meaning the order in which matrices are multiplied matters: A*B is not necessarily the same as B*A.

When applied to our system of linear equations, matrix multiplication is used to determine the unknown variables. After calculating the inverse matrix A^{-1}, we multiply it by vector B, yielding our desired vector of currents I. This operation requires careful attention, as each element of the resulting matrix is the sum of the products of corresponding elements from the rows of A^{-1} and the columns of B. The resulting vector I provides the values for the currents through the different paths in the circuit as dictated by Kirchhoff's Law.