A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.

Question

Quizplus · Accepted Answer

We can write the given system of equations in matrix form as AX=B, where
A = [10 -3 2; 2 -1 1; 1 1 1]
X = [x; y; z]
B = [7; 1; 2]
To find the general solution, we need to find the inverse of matrix A. Using a calculator or Gaussian elimination, we can find that
A^(-1) = [1/4 3/4 -5/4; -1/2 -1/2 3/2; 1/4 1/4 -1/4]
Now, we can write the solution as X = A^(-1)B + t[U], where t is any real number and U is the column matrix [u; v; w] representing the general solution. So, the solution is
X = [1/4 3/4 -5/4; -1/2 -1/2 3/2; 1/4 1/4 -1/4][7; 1; 2] + t[u; v; w]
Simplifying this expression, we get
X = [3-t/4+u/4; -1/2-t/2+v/4; 1-t/4+w/4]
So, the general solution is
x = 3 - t/4 + u/4
y = -1/2 - t/2 + v/4
z = 1 - t/4 + w/4
where t, u, v, and w are any real numbers. Therefore, the system has an infinite number of solutions.

A System of Equations May Have a Unique Solution, an Infinite