How to make a matrix for this circuit to find the currents

Question

I have this circuit (https://www2.0zz0.com/2025/02/25/00/383457466.png) from my homework, it has all the values but I will write all of them here.

R1 = 6Ω

R2 = 3Ω

R3 = 2Ω

R4 = 2Ω

V1 = 14V

V2 = 5V

My teacher wants us to make a matrix for this circuit to use it on (https://www.desmos.com/matrix), so we can get the currents. However, I don't know how to make the matrix and don't understand how to loops work.

Answer 1

Draw 3 clockwise current loops and label them I_a, I_b, and I_c.

I_a will go from the 14-Volt source and oppose (flowing + to −)

the polarities of 6Ω & 3Ω. With I_b flowing through 6Ω in the

opposite direction and with I_c completely

removed from the I_a loop, the first equation will be written as

I_a: (6Ω + 3Ω)I_a − 6ΩI_b + 0ΩI_c = 14 V.

Next, I_b will flow opposite (from + to −) the polarities of 6Ω, 5V, and 2Ω

with I_a flowing through 6Ω against I_b and with I_c having no influence on the

second loop. Then write the second equation as

I_b: (2Ω + 6Ω)I_b − 6ΩI_a + 0ΩI_c = -5V.

For the final equation, I_c will flow from 5V against the polarity (flowing + to −)

of 2Ω at the right end of the circuit and with I_a and I_b having no influence on the

third loop. The third equation is then written as

I_c: 2ΩI_c + 0ΩI_a +0ΩI_b + = 5V.

Write the coefficients of I_a, I_b, and I_c in each equation in a-b-c order:

9--------(-6)--------0

(-6)-------8---------0 •

0----------0---------2

This last matrix multiplied by the column matrix

[I_a

I_b

I_c]

is equated to the column matrix

[14

-5

5]

to give (via Matrix Algebra and Cramer's Rule)

I_a = (41/18) Ampères [or A]

I_b = (13/12) A

I_c = (5/2) A.

From inspection of the given circuit diagram and the values of

I_a, I_b, and I_c found above, I would submit:

I₁ = I_a − I_b or (43/36) A;

I₂ = I_a or (41/18) A;

I₃ = I_b or (13/12) A;

I₄ = I_c = (5/2) A;

I₅ = I_c − I_b or (17/12) A.