4x + y + z = 6

-x +y - 4z =-19

x + 3y + 4z = 11

4x + y + z = 6 | 4 1 1 || x | = | 6|

-x + y – 4z = -19 |-1 1 -4 || y | = |-19|

x + 3y + 4z = 11 | 1 3 4|| z | = | 11|

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, [6 -19 11].

M = | 4 1 1| M^T = | 4 -1 1|

|-1 1 -4| | 1 1 3|

| 1 3 4| | 1 -4 4|

Adj(M) = cofactors of M^T.(Or, the transpose of the cofactors of M – same thing)

Adj(M) = |1 3| = 4 + 12 |1 3| = 4 – 3 |1 1|= -4 – 1

|-4 4| = __16__ |1 4| = __1__ |1 -4| = __-5__

|-1 1| = -4 + 4 |4 1| = 16 – 1 |4 -1| = -16 + 1

|-4 4| __ 0__ |1 4| = __15__ |1 -4| = -1__5__

|-1 1| = -3 - 1 |4 1| = 12 - 1 |4 -1| = 4 + 1

| 1 3| = __-4__ |1 3| = __11__ |1 1| = __5__

Adj(M) = |16 1 -5| x |+ - +| = |16 -1 -5|

| 0 15 -15| |- + - | | 0 15 15|

|-4 11 5| |+ - +| |-4 -11 5|

det(M) = 4|1 -4| - 1|-1 -4| + 1|-1 1| = 4(4 + 12) – 1(-4 +4) + 1(-3 – 1) = 64 + 0 – 4 =__ 60__

|3 4| | 1 4| | 1 3|

__det(M) = 60__

__Inverse Matrix__

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

M^(-1) = (1/60) * |16 -1 -5|

| 0 15 15|

|-4 -11 5|

X = M^(-1) * R

X = (1/60)*|16 -1 -5| * | 6| = (1/60) * | 96 + 19 – 55| = (1/60)*| 60| =| 1 |

| 0 15 15| |-19| | 0 – 285 + 165| |-120| |-2|

|-4 -11 5| | 11| |-24 + 209 + 55| | 240| | 4|

__Solution: x = 1, y = -2, z = 4__