Question

KVL equations of circuit can be written as

10I₁ − 2I₂ − 5I₃ = 25
−2I₁ + 13I₂ − 1I₃ = 2
−5I₁ − I₂ + 10I₃ = 15
​
Write is in form of A.X=b and solve it with LU decomposition method
Decompose the coefficient matrix as L and U

Answer 1

The question is about solving a set of linear equations using LU decomposition, where the equations are derived from Kirchhoff's Voltage Law in a circuit analysis problem. The solution involves expressing the equations in matrix form, decomposing the matrix, and then solving for the unknown currents through forward and backward substitution.

Step-by-step explanation:

The question deals with solving a set of linear equations derived from Kirchhoff's Voltage Law (KVL) for a circuit, using the LU decomposition method. The given equations are:

1. 10I₁ − 2I₂ − 5I₃ = 25
2. − 2I₁ + 13I₂ − 1I₃ = 2
3. − 5I₁ − I₂ + 10I₃ = 15

We can express these equations in matrix form A.X = b, where A is the coefficient matrix, X is the variable matrix, and b is the constant matrix. To find the solution, we decompose matrix A into an upper triangular matrix (U) and a lower triangular matrix (L), such that A = L.U. The system is then solved in two steps:

1. Solve L.y = b for y using forward substitution.
2. Solve U.x = y for x using backward substitution.

After decomposition and substitution, we find the values of currents I1, I2, and I3.