Find (x ,y) such that (x, y, 1, -2) are coefficients for a dependence relation for m1, m2, m3, m4.

Question

Let

m₁ = (1 1) m₂ =(1 2) m₃ = (1 3) m₄ = (3 2)

1 1 2 3 3 5 2 1

Find (x ,y) such that (x, y, 1, -2) is a dependence relation for m₁, m₂, m₃, m₄.

a. (-9,4)

b. (-4,9)

c. (4,-9)

d. (9,-4)

Answer 1

xm₁ + ym₂ + m₃ - 2m₄ = the 2 x 2 zero matrix

Equating the entries in the first row and first column, we get x + y + 1 - 6 = 0

Equating the entries in the first row second column, we get x + 2y + 3 - 4 = 0

So, we have the system of equations:

x + y = 5

x + 2y = 1

Subtract the equations to obtain -y = 4. So, y = -4 and x = 9.