X+2y-2z=-11 2x +z =-6 5y -3z = -7 using a matrix: 1 2 -2| -11 2 0 1 | -7 0 5 -3 | -6

x + 2y – 2z = -11 | 1 2 -2|| x | = | -11|

2x + z = -6 | 2 0 1 || y | = | -6|

5y – 3z = -7 | 0 5 -3|| z | = | -7|

Starting with MX = R, we find the inverse of M, M^(-1), using which we evaluate the unknowns matrix, X, with the matrix equation, X = M^(-1) * R, where R is the constant matrix, [-11 -6 -7].

M = | 1 2 -2| M^T = | 1 2 0|

| 2 0 1| | 2 0 5|

| 0 5 -3| |-2 1 -3|

Adj(M) = |0 5| = 0 – 5 | 2 5| = -6 + 10 | 2 0| = 2 – 0

|1 -3| = __-5__ |-2 -3| = __4__ |-2 1| = __2__

|2 0| = -6 – 0 | 1 0| = -3 – 0 | 1 2| = 1 + 4

|1 -3| __ -6__ |-2 -3| = __-3__ |-2 1| = __5__

|2 0| = __10__ | 1 0| = __5__ | 1 2| = __-4__

|0 5| | 2 5| | 2 0|

Adj(M) = |-5 4 2| x |+ - +| = |-5 -4 2|

|-6 -3 5| |- + -| | 6 -3 -5|

|10 5 -4| |+ - +| |10 -5 -4|

det(M) = 1|0 1| - 2|2 1| - 2|2 0| = (0 – 5) – 2(-6 – 0) – 2(10 – 0) = -5 + 12 – 20 =__ -13__

|5 -3| |0 -3| |0 5|

__det(M) = -13__

__Inverse Matrix__

M^(-1) = 1/det(M) * Adj(M)

M^(-1) = (-1/13) * |-5 -4 2|

| 6 -3 -5|

|10 -5 -4|

X = M^(-1) * R

X = (-1/13)*|-5 -4 2| * | -11 | = (-1/13) * | 55 + 24 – 14| = (-1/13)*| 65| = |-5 |

| 6 -3 -5| | -6 | |-66 + 18 + 35| |-13| | 1 |

|10 -5 -4| | -7 | |-110 + 30 + 28| |-52| | 4 |

__Solution: x = -5, y = 1, z = 4__