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| = -15
|-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