Solve the following system of equations by using matrix inverse method.
3x+4y+7z=14
2x-y+3z=4
2x+2y-3z=0
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Answers
1.) write the equations in matrix form.
2.) The matrix is the form of AX = B
Where A is the matrix if the unknowns, X the unknowns and be the solutions in the equations.
3.) Get the inverse of the matrix.
X = A⁻¹B for values of the unknowns.
Getting an inverse of a 3x3 matrix involves the following steps :
A.) Get the determinant.
B.) Transpose the original matrix.
C.) Finding the determinant of each of the 2 × 2 minor matrices.
D.) Getting matrix of cofactors.
E.) Getting adjoin of matrix.
F.) Dividing the adjoin by the determinant to find the inverse.
Find calculations attached in the image.
Answer:
Step-by-step explanation:
We can write the system of equations in matrix form as:
| 3 4 7 | | x | | 14 |
| 2 -1 3 | x | y | = | 4 |
| 2 2 -3 | | z | | 0 |
We can solve for x, y, and z by using matrix inverse method:
Step 1: Write the augmented matrix [A | B] where A is the coefficient matrix and B is the constants matrix:
| 3 4 7 | 14 |
| 2 -1 3 | 4 |
| 2 2 -3 | 0 |
Step 2: Find the inverse of matrix A.
We can use the formula A^-1 = 1/det(A) * adj(A), where det(A) is the determinant of matrix A and adj(A) is the adjugate matrix of A.
det(A) = 3(-1)(-3) + 4(3)(2) + 7(2)(2) - 7(-1)(2) - 4(2)(-3) - 3(3)(4) = 121
adj(A) = | (-1)(-3) 3(-3) 3(2) |
| 4(-2) (-1)(-3) 4(3) |
| 4(7) 2(2) (-1)(4) |
= | 9 -9 6 |
| 8 3 12 |
| 28 -4 -8 |
A^-1 = (1/121) * | 9 -9 6 | = | 9/121 -9/121 6/121 |
| 8 3 12 | | 8/121 3/121 12/121 |
| 28 -4 -8 | |28/121 -4/121 -8/121 |
Step 3: Multiply A^-1 by B to get the solution vector [x y z].
| 9/121 -9/121 6/121 | | 14 | | 1 |
| 8/121 3/121 12/121 | x | 4 | = | 1 |
|28/121 -4/121 -8/121 | | 0 | | -1 |
Therefore, the solution to the system of equations
is x=1, y=1, and z=-1.
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