Final answer:
To solve the given system of equations using an inverse matrix, we first convert the system into matrix form, calculate the inverse of the coefficient matrix, and then multiply this inverse with the constants matrix. The solution is x=-4.4 and y=-1.3, using the inverse matrix A⁻¹ = [[0.4, 1],[0.3, 0.5]].
Step-by-step explanation:
To solve the system of equations using an inverse matrix, let's first write the equations in matrix form. The system:
-5x + 10y = 9
3x - 4y = -8
Can be written as:
AX = B
Where A is the coefficient matrix, X is the variable matrix, and B is the constants matrix:
A = [[-5, 10],
[3, -4]]
X = [[x],
[y]]
B = [[9],
[-8]]
Now we need to find the inverse matrix A⁻¹, and use it to solve for X:
A⁻¹ * A * X = A⁻¹ * B
X = A⁻¹ * B
The inverse of A (A⁻¹) is calculated as:
A⁻¹ = 1/det(A) * adj(A)
Where det(A) is the determinant of A and adj(A) is the adjugate of A. The determinant is calculated as:
(-5)(-4) - (3)(10) = 20 - 30 = -10
And the adjugate matrix (switch the positions of a and d, change the signs of b and c) is:
[[ -4, -10],
[ -3, -5]]
So the inverse matrix A⁻¹ is:
A⁻¹ = 1/(-10) * [[ -4, -10],
[ -3, -5]]
= [[0.4, 1],
[0.3, 0.5]]
Now, we multiply the inverse matrix A⁻¹ by B to get X:
X = [[0.4, 1],
[0.3, 0.5]] * [[9],
[-8]]
= [[(0.4)(9) + (1)(-8)],
[(0.3)(9) + (0.5)(-8)]]
= [[3.6 - 8],
[2.7 - 4]]
= [[-4.4],
[-1.3]]
Therefore, the solution to the system is x=-4.4 and y=-1.3, and the inverse matrix used was:
A⁻¹ = [[0.4, 1],
[0.3, 0.5]]