Ques.- Solve the following system of linear equation by matrix method (inverse method) .
12p + 4q + 17r +11s = 5
14p + 8q + 9r + 4s = 8
3p + 7q +2r + 5s = 12
5p + 12q +9r +7s = 7
Rather tedious to find the inverse of a 4by4 matrix, and out of the question to type it all out.
Surely you can find an inverse calculator on the web.
Just google and pick one
then multiply that inverse by
5
8
12
7
To solve the given system of linear equations using the matrix method, we will create a matrix equation and then find the inverse of the coefficient matrix.
Step 1: Create a matrix equation
The given system of linear equations can be represented in matrix form as follows:
| 12 4 17 11 | | p | | 5 |
| 14 8 9 4 | * | q | = | 8 |
| 3 7 2 5 | | r | | 12 |
| 5 12 9 7 | | s | | 7 |
Step 2: Find the inverse of the coefficient matrix
To find the inverse of the coefficient matrix, we can use the formula:
A^(-1) = (1/det(A)) * adj(A)
First, let's find the determinant (det(A)) of the coefficient matrix:
det(A) = 12(8 * 9 - 4 * 2) - 4(14 * 9 - 3 * 2) + 17(14 * 7 - 3 * 12) - 11(14 * 5 - 3 * 12)
= 12(72 - 8) - 4(126 - 6) + 17(98 - 36) - 11(70 - 36)
= 12(64) - 4(120) + 17(62) - 11(34)
= 768 - 480 + 1054 - 374
= 968
Next, let's find the adjoint (adj(A)) of the coefficient matrix:
adj(A) = | (8 * 9 - 4 * 2) -(14 * 9 - 3 * 2) (14 * 7 - 3 * 12) -(14 * 5 - 3 * 12) |
| -(4 * 17 - 12 * 2) (12 * 9 - 3 * 17) -(12 * 7 - 2 * 17) (12 * 5 - 17 * 2) |
| (4 * 17 - 12 * 8) -(12 * 14 - 4 * 17) (12 * 14 - 2 * 7) (12 * 17 - 7 * 4) |
| -(4 * 11 - 17 * 2) (12 * 11 - 14 * 2) -(14 * 2 - 3 * 11) (14 * 17 - 12 * 11) |
Calculating these values, we get:
adj(A) = | 64 -108 -30 30 |
| -76 57 6 58 |
| -16 62 -56 29 |
| -34 26 33 92 |
Now, we can calculate A^(-1):
A^(-1) = (1/det(A)) * adj(A)
= (1/968) * | 64 -108 -30 30 |
| -76 57 6 58 |
| -16 62 -56 29 |
| -34 26 33 92 |
Step 3: Calculate the solution vector
Finally, we can calculate the solution vector by multiplying the inverse of the coefficient matrix (A^(-1)) with the constant vector on the right-hand side of the matrix equation.
| p | | 5 | | 64 -108 -30 30 | | 5 |
| q | = | 8 | * | -76 57 6 58 | * | 8 |
| r | | 12 | | -16 62 -56 29 | | 12 |
| s | | 7 | | -34 26 33 92 | | 7 |
Performing the matrix multiplication, we get:
| p | | (-147/242) |
| q | = | (-176/121) |
| r | | (825/484) |
| s | | (221/242) |
Therefore, the solution to the given system of linear equations is:
p = -147/242
q = -176/121
r = 825/484
s = 221/242