Home > Precalculus > Solving a System of Equations Using a Matrix

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

Answer 1

#x=1 and y=2#

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#

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