Which of the following is the solution to the matrix equation below?

[0 9 2] [18]
[3 2 1] x=[5]
[1 -1 0] [-1]

a) [-3 -2 18]

b) [-3 2 -18]

c) [-3]
[-2]
[18]

d)[-3]
[2]
[-18]

To find the solution to the matrix equation, you need to solve for the values of x, y, and z in the equation Ax = b, where A is the given matrix, x is the unknown vector, and b is the right-hand side vector.

To solve this equation, you can use the inverse of matrix A. If A is invertible, the solution to the equation is given by x = A^(-1)b.

First, you need to calculate the inverse of matrix A. If A is a 3x3 matrix [a b c; d e f; g h i], the formula to calculate its inverse is:

A^(-1) = (1/det(A)) * adj(A)

Where det(A) is the determinant of matrix A, and adj(A) is the adjugate of A.

Calculating the determinant:
det(A) = 0(2*0 - (-1)*(-1)) - 9(3*0 - (-1)*1) + 2(3*(-1) - 2*1)
= 0 - 9(-1) + 2(-5)
= -9 + (-10)
= -19

Calculating the adjugate:
adj(A) = [e*i - f*h, -(d*i - f*g), d*h - e*g;
-(b*i - c*h), a*i - c*g, -(a*h - b*g);
b*f - c*e, -(a*f - c*d), a*e - b*d]
= [2*0 - 1*(-1), -((3*0 - 1*1)), 3*(-1) - 2*1;
- (9*0 - 1*(-1)), 0*0 - 1*2, - (0*(-1) - 1*3);
9*(-1) - 2*0, - (0*(-1) - 9*3), 0*0 - 9*2]
= [1, -1, -5;
1, -2, 3;
-9, -27, 0]

Now, you can calculate the inverse of matrix A:
A^(-1) = (1/det(A)) * adj(A)
= (-1/19) * [1, -1, -5;
1, -2, 3;
-9, -27, 0]
= [-(1/19), 1/19, 5/19;
-(1/19), 2/19, -3/19;
9/19, 27/19, 0]

Next, multiply the inverse of matrix A by the vector b to find the solution x:
x = A^(-1)b
= [-(1/19), 1/19, 5/19; -(1/19), 2/19, -3/19; 9/19, 27/19, 0] * [18; 5; -1]
= [(-1/19)*18 + (1/19)*5 + (5/19)*(-1); (-(1/19))*18 + (2/19)*5 + (-3/19)*(-1); (9/19)*18 + (27/19)*5 + 0*(-1)]
= [-3; 2; -18]

Therefore, the solution to the matrix equation is x = [-3; 2; -18].

The correct answer from the given options is d) [-3; 2; -18].