Lawrence A. answered • 03/14/19
Patient and experienced tutor
There are many methods for solving matrix equations, such as the Gauss Elimination method.
Because the matrix is not large (3x3), we shall use Cramer's rule.
In matrix form, the equation is
| 2 3 -4 | | x| | -4 |
| 3 -2 5 | | y | = | 14 |
| 5 1 -3 | | z | | -2 |
This equation is of the form [A]{x} = {b}
where [A] is the 3x3 matrix, {x} represents {x,y,z} and {b} represents the right hand side(RHS).
The determinant of [A] is
D = det(A) = 2(6 - 5) - 3(-9 - 25) - 4(3 + 10) = 2 + 102 - 52 = 52
To solve for x, replace the 1st column of [A] by{b}, the RHS. Evaluate the determinant and divide by D..
The determinant to be evaluated is of the matrix
| -4 3 -4 |
| 14 -2 5 |
| -2 1 - 3 |
Evaluate its determinant to obtain 52.
Therefore, x = 52/D = 52/52 = 1.
Next, replace the 2nd column of [A] by {b} and evaluate the determinant. The solution for y is this determinant of this matrix divided by D.
The matrix is
| 2 -4 -4 |
| 3 14 5 |
} 5 -2 -3 |
and its determinant is 104.
Therefore, y = 104/D = 104/52 = 2.
Next, replace the 3rd column of [A] by {b}, and evaluate the determinant of the matrix. This yields
| 2 3 -4 |
| 3 -2 14 |
| 5 1 -2 |
whose determinant is 156.
Therefore, z = 156/D = 156/52 = 3.
The solution for the matrix equation is
x = 1, y = 2, z = 3.
I hope this helps.
Another popular method is Gauss Elimination, followed by Back Substitution. Contact me if you want to use this method.
