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Solve the following set of equations by Matrix inversion method
3x + 2y - 5z = 12
3y +4z - 2 = 0
z -x +2y = 2?
3x + 2y - 5z = 12
3y +4z - 2 = 0
z -x +2y = 2?
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Solve the following set of equations by Matrix inversion method 3x + 2...
Matrix Inversion Method to Solve a Set of Equations
Step 1: Write the Equations in Matrix Form
Given equations:
3x + 2y - 5z = 12
3y + 4z - 2 = 0
z - x + 2y = 2
Rewriting the equations in matrix form:
AX = B
Where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
A =
| 3 2 -5 |
| 0 3 4 |
| -1 2 1 |
X =
| x |
| y |
| z |
B =
| 12 |
| 2 |
| 2 |
Step 2: Find the Inverse of Matrix A
To solve for X, we need to find the inverse of matrix A:
A^-1 = 1/det(A) * adj(A)
Where det(A) is the determinant of A and adj(A) is the adjugate of A.
Step 3: Calculate the Determinant and Adjugate of Matrix A
det(A) = 3(3*1 - 2*4) - 2(0*1 - 4*(-1)) + (-5)(0*2 - 3*(-1))
det(A) = 3(3 - 8) - 2(0 + 4) + (-5)(0 + 3)
det(A) = 3(-5) - 2(4) - 5(3)
det(A) = -15 - 8 - 15
det(A) = -38
adj(A) =
| 3 4 11 |
| 5 -5 -3 |
| -2 3 -6 |
Step 4: Find the Inverse of Matrix A
A^-1 = 1/-38 * adj(A)
A^-1 =
| -3/38 -4/38 -11/38 |
| -5/38 5/38 3/38 |
| 2/38 -3/38 6/38 |
Step 5: Solve for X
X = A^-1 * B
X =
| -3/38 -4/38 -11/38 | | 12 |
| -5/38 5/38 3/38 | * | 2 |
| 2/38 -3/38 6/38 | | 2 |
Solving the matrix multiplication, we get the values of x, y, and z.
Therefore, x = -2, y = 3, z = 4.
This is how the set of equations is solved using the Matrix Inversion method.
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Solve the following set of equations by Matrix inversion method 3x + 2y - 5z = 12 3y +4z - 2 = 0 z -x +2y = 2?
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