solve the set of linear equations by the matrices method: a+3b+2c=3, 2a-b-3c=-8,5a+2b+c=9
determinant of coefficient matrix
1 3 2
2 -1 -3
5 2 1
is -28
now replace the first column with the the constants coefficients
3 3 2
-8 -1 -3
9 2 1
and the determinant of that is -56
do a = -56/-28 = 2
now replace the 2nd column with the constants and find the determinant to get b = 84/-28= -3
and similarly replace the 3rd column with the constants to get c = -140/-28 = 5
a = 2
b = -3
c = 5
I used a matrix method called Cramer's Rule
Khan as usual has a nice simple clip explaining how to find the determinant of a 3 by 3 matrix
https://www.khanacademy.org/math/precalculus/precalc-matrices/inverting_matrices/v/finding-the-determinant-of-a-3x3-matrix-method-1
you might also click on his method-2 of the same thing
In google type :
Systems Matrix Calculator
When you see list of result click on :
Online calculator. Linear equations solver: Matrix solution.
When page be open type your coefficients
1 , 3 , 2 , 3
2 , - 1 , - 3 , - 8
5 , 2 , 1 , 9
and click opštion :
Solve the system.
You will see solution step -by step.
Remark :
x1 = a , x2 = b , x3 = c
To solve the given set of linear equations using the matrix method, we will represent the coefficients of the variables as a matrix, the variables as a column matrix, and the constants as another column matrix.
Let's denote the coefficient matrix as A, the variable matrix as X, and the constant matrix as B.
So, the coefficient matrix A is:
A = [[1, 3, 2], [2, -1, -3], [5, 2, 1]]
The variable matrix X is:
X = [[a], [b], [c]]
The constant matrix B is:
B = [[3], [-8], [9]]
We can then represent the given set of equations in matrix form as follows:
AX = B
To find X, we need to solve the equation AX = B for X.
The inverse of matrix A (A^-1) can be computed using matrix operations such as determinant, cofactor, and adjugate. However, rather than manually calculating the inverse, we can use the numpy library in Python, which provides a ready-made function to calculate the inverse of a matrix.
Here's the code to solve the equations using the matrix method in Python:
import numpy as np
A = np.array([[1, 3, 2], [2, -1, -3], [5, 2, 1]])
B = np.array([[3], [-8], [9]])
# Calculate the inverse of matrix A
A_inv = np.linalg.inv(A)
# Solve for X
X = np.dot(A_inv, B)
print("The solution to the system of equations is:")
print("a =", X[0][0])
print("b =", X[1][0])
print("c =", X[2][0])
When you run this code, you will get the values of a, b, and c, which are the solution to the given set of linear equations.