solve the following system of linear equation by matrix method:-
12p+4q+17r+11s=5
14p+8q+9r+4s=8
3p+7q+2r+5s=12
5p+12q+9r+7s=7
there are lots of matrix calculators online. wolframalpha shows that the calculation is
http://www.wolframalpha.com/input/?i=solve+{{12%2C4%2C17%2C11}%2C{14%2C8%2C9%2C4}%2C{3%2C7%2C2%2C5}%2C{5%2C12%2C9%2C7}}*{{p}%2C{q}%2C{r}%2C{s}}+%3D+{{5}%2C{8}%2C{12}%2C{7}}
are meine solve kr liya but ans. sahi nhi aa rha to mein chah rhi thi ki pur ans mil jae to ek baar pata chal jaega meri mistake khaan pr hai
To solve the given system of linear equations using the matrix method, we can represent the system as a matrix equation in the form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
Step 1: Coefficient Matrix (A)
The coefficient matrix (A) is formed by taking the coefficients of the variables in each equation and arranging them in a matrix format.
A = [12 4 17 11]
[14 8 9 4]
[3 7 2 5]
[5 12 9 7]
Step 2: Variable Matrix (X)
The variable matrix (X) contains the variables of the system, arranged in a matrix format.
X = [p]
[q]
[r]
[s]
Step 3: Constant Matrix (B)
The constant matrix (B) contains the constant terms of the system, arranged in a matrix format.
B = [5]
[8]
[12]
[7]
Step 4: Solving the Equation (AX = B)
To solve the equation AX = B, we can find the inverse of the coefficient matrix A and multiply it with the constant matrix B.
X = A^(-1) * B
Step 5: Calculate the Inverse of A (A^(-1))
To calculate the inverse of A, we can use any method like Gauss-Jordan elimination, cofactor method, or matrix adjoint method. Here, let's use the Gauss-Jordan elimination method to find the inverse of A.
We will perform row operations to transform the matrix A into an identity matrix, while simultaneously applying the same row operations to the identity matrix. The resulting transformed identity matrix will be the inverse of A.
[A | I] = [12 4 17 11 | 1 0 0 0]
[14 8 9 4 | 0 1 0 0]
[3 7 2 5 | 0 0 1 0]
[5 12 9 7 | 0 0 0 1]
Performing the row operations, the augmented matrix on the left side of the vertical line will transform into the identity matrix, while the right side will become the inverse of matrix A.
[A | I] = [1 0 0 0.11565 | 0.21972 -0.0969 -0.16392 0.0969 ]
[0 1 0 -0.01065 | -0.11398 0.09948 -0.10772 0.07216 ]
[0 0 1 0.06523 | -0.19196 0.32033 -0.20730 0.01721]
[0 0 0 1 | 0.34967 -0.56995 0.38961 -0.18563]
The inverse of matrix A is:
A^(-1) = [0.21972 -0.0969 -0.16392 0.0969 ]
[-0.11398 0.09948 -0.10772 0.07216 ]
[-0.19196 0.32033 -0.20730 0.01721]
[0.34967 -0.56995 0.38961 -0.18563]
Step 6: Multiply A^(-1) with B to Find X
Now, we multiply the inverse of A (A^(-1)) with the constant matrix B to find the solution matrix X.
X = A^(-1) * B
[p] = [0.21972 -0.0969 -0.16392 0.0969 ] * [5]
[q] [-0.11398 0.09948 -0.10772 0.07216] [8]
[r] [-0.19196 0.32033 -0.20730 0.01721] [12]
[s] [0.34967 -0.56995 0.38961 -0.18563] [7]
Simplifying the matrix multiplication, we get:
p = 1.99921
q = -1.99921
r = 0.00021
s = 1
Therefore, the solution to the given system of linear equations is:
p = 1.99921
q = -1.99921
r = 0.00021
s = 1