Question

Solve the following system of equations by using the inverse of the coefficient matrix.

7x?y+4z=?3

?3y+8z=?20

-2x+4y+5z=-42

Answer 1

The given system of linear equations can be written in matrix form as AX = b, where b = (-3,-20,-42)^T, X = (x, y, z)^T and the coefficient matrix A =

7	-1	4
0	-3	8
-2	4	5

Now, let M = [A|I₃] =

7	-1	4	1	0	0
0	-3	8	0	1	0
-2	4	5	0	0	1

The RREF of M is

1	0	0	47/337	-21/337	-4/337
0	1	0	16/337	-43/337	56/337
0	0	1	6/337	26/337	21/337

Hence A^-1 =

47/337	-21/337	-4/337
16/337	-43/337	56/337
6/337	26/337	21/337

Further, since AX = b, hence, on multiplying to the left by A^-1, we get A^-1AX = A^-1b or, = A^-1b = (447/337,-1540/337,-1420/337)^T. Hence, x = 447/337,y = -1540/337 and z = -1420/337.