Question

If P = `[(1,-1,0),(2,3,4),(0,1,2)]` and Q = `[(2,2,-4),(-4,2,-4),(2,-1,5)]` find (QP) and hence solve the following system of equations using matrices:

x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7

Answer 1

Given,

QP = `[(1,-1,0),(2,3,4),(0,1,2)] [(2,2,-4),(-4,2,-4),(2,-1,5)]`

= `[(6,0,0),(0,6,0),(0,0,6)]`

= `6[(1,0,0),(0,1,0),(0,0,1)]`

= 6I

By the system of equations:

A = `[(1,-1,0),(2,3,4),(0,1,2)];X=[(x),(y),(z)];B=[(3),(17),(7)]`

AX = B

X = A^–1B

QA = 6I

Post multipication of A^–1 both sides,

QAA^–1 = 6IA^–1     ...[∵ AA^–1 = I]

Q = 6A^–1 

A^–1 = `1/6`Q

X = A^–1B

= `1/6`QB

A^–1 = `1/6[(2,2,-4),(-4,2,-4),(2,-1,5)][(3),(17),(7)]`

X = `1/6[(6+34-28),(-12+34-28),(6-3+35)]`

X = `1/6[(12),(-6),(24)]`

`[(x),(y),(z)]=[(2),(-1),(4)]`

Hence, x = 2, y = –1, z = 4