How do you solve x - 3y = -5 and -x+ y = 1 using matrices?

1 Answer
Aug 24, 2016

x=1 and y=2

Explanation:

We can express this by splitting the equations up into a coefficient matrix with a vector for the variables and a vector for the solutions.

Avec(u) = vec(v)

where A is the coefficient matrix. Assuming A is invertible we can left multiply both sides of the equation by the inverse of A, denoted A^(-1) to obtain

Ivec(u) = A^(-1)vec(v)

where I is the n x n identity matrix.

For a 2xx2 matrix

A = ((a,b),(c,d))

The inverse of A is given by:

A^(-1) = (1)/(ad-bc)((d,-b),(-c,a))

In the context of this problem we have

((1,-3),(-1,1))((x),(y)) = ((-5),(1))

so A = ((1,-3),(-1,1))

A^(-1) = -1/2((1,3),(1,1))

Just to check that this is correct:

A^(-1)A = -1/2((1,3),(1,1))((1,-3),(-1,1))

=-1/2((-2,0),(0,-2)) = ((1,0),(0,1)) as required

so we have:

((x),(y)) = -1/2((1,3),(1,1)) ((-5),(1))

((x),(y)) = -1/2((-2),(-4)) = ((1),(2))

therefore x = 1 and y = 2