Solve the system by inverting the coefficient matrix and using the following theorem:

Question

If A is an invertible n x n matrix, then for each n x 1 matrix b, the system of equations A x = b has exactly one solution, namely, x = a^-1b.

x + y + z = 3

x + y - 2z = 6

-2x + y + z = 0

Answer 1

Turn the system of equations into 2 matrices. The first is the coefficients of the "x", "y", and "z" variables (everything left of the equal sign) like this:

The 2nd matrix is the constant terms (everything right of the equal sign) like this:

Then take the inverse of matrix A. There are manual methods but calculators like the TI-84 do this automatically. Just put the matrix in the calculator, for instance as matrix A. And then do A^-1 to get the inverse. In this case it is:

Then multiply A^-1 and B like A^-1 • B and you'll get the answer:

Meaning x = 1, y = 3, and z = -1