solve the system of linear equations 4x-6y=-2 , x+3y=7 by using matrix inversion method. solve that question and also these
To solve the system of linear equations:
1) Write the system in matrix form: AX = B
where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
The given system:
4x - 6y = -2 ----(1)
x + 3y = 7 ----(2)
Matrix form:
A = [4 -6]
[1 3]
X = [x]
[y]
B = [-2]
[7]
2) Calculate the inverse of matrix A, if it exists.
To find the inverse of A:
1) Calculate the determinant of A, denoted as |A|.
|A| = 4(3) - (-6)(1) = 18 + 6 = 24
2) Swap the positions of the diagonal elements and change the sign of the off-diagonal elements.
A^(-1) = [ 3/24 -(-6)/24 ]
[ -1/24 4/24 ]
Simplifying A^(-1):
A^(-1) = [1/8 1/4 ]
[ -1/24 1/6 ]
3) Multiply the inverse of A by B to solve for X:
X = A^(-1)*B
X = [1/8 1/4 ] * [ -2 ]
[ -1/24 1/6 ] [7 ]
Performing the matrix multiplication:
X = [ (1/8)(-2) + (1/4)(7) ]
[ (-1/24)(-2) + (1/6)(7) ]
Simplifying:
X = [ (-2/8) + (7/4) ]
[ (2/24) + (7/6) ]
X = [ (-1/4) + (7/4) ]
[ (2/24) + (14/24) ]
X = [ 6/4 ]
[ 16/24 ]
Simplifying X:
X = [ 3/2 ]
[ 2/3 ]
Therefore, the solution to the system of linear equations is:
x = 3/2
y = 2/3
Similarly, you can follow the steps to solve the other systems of linear equations using matrix inversion method.
To solve a system of linear equations using matrix inversion method, we need to convert the system into matrix form.
The given system of equations is:
4x - 6y = -2 ----(1)
x + 3y = 7 ----(2)
To represent this system in matrix form, we can write it as AX = B, where:
A = [[4, -6], [1, 3]]
X = [[x], [y]]
B = [[-2], [7]]
To find the solution, we can use the formula:
X = A^(-1) * B
First, let's find the inverse of matrix A.
Step 1: Find the determinant of matrix A:
det(A) = (4 * 3) - (-6 * 1) = 12 + 6 = 18
Step 2: Find the adjoint of matrix A:
The adjoint of A is obtained by taking the transpose of the matrix of cofactors.
The cofactor matrix of A is obtained by taking the determinant of each minor in A.
Cofactor of 4: -3 (Switch the signs alternately for each element)
Cofactor of -6: -1
Cofactor of 1: 6
Cofactor of 3: 4
So, the adjoint of matrix A is:
adj(A) = [[-3, 6], [-1, 4]]
Step 3: Find the inverse of matrix A:
A^(-1) = (1/det(A)) * adj(A)
= (1/18) * [[-3, 6], [-1, 4]]
= [[-1/6, 1/3], [-1/18, 2/9]]
Now, let's calculate X using the inverse of matrix A and B.
X = A^(-1) * B
= [[-1/6, 1/3], [-1/18, 2/9]] * [[-2], [7]]
= [[(-1/6) * (-2) + (1/3) * 7], [(-1/18) * (-2) + (2/9) * 7]]
= [[1 + 7/3], [1/9 + 14/9]]
= [[10/3], [15/9]]
= [[10/3], [5/3]]
So, the solution to the given system of equations is:
x = 10/3
y = 5/3
Let me know if there's anything else I can help you with.