Find the matrix X such that XA=B ..?

A = ((3a, 2b),(-a,b))

Above is a single matrix A shown.

B = (( -a , b ),(2a, 2b))

Above shown is a single matrix B.

1 Answer
May 29, 2017

X = [(0,1),(4/5,2/5)]

Wolfram Alpha confirms this.


Here is another way to do this. We write

XA = B,

and

[(x_1,x_2),(x_3,x_4)][(3a,2b),(-a,b)] = [(-a,b),(2a,2b)].

From matrix multiplication, we obtain the following system of equations:

3ax_1-ax_2 = a(3x_1 - x_2) = -a " "" "bb((1))

2bx_1 + bx_2 = b(2x_1 + x_2) = b " "" "" "bb((2))

3ax_3 - ax_4 = a(3x_3 - x_4) = 2a " "" "" "bb((3))

2bx_3 + bx_4 = b(2x_3 + x_4) = 2b " "" "" "bb((4))

From (1), we have:

3x_1 - x_2 = -1

=> x_1 = x_2/3 - 1/3

From plugging x_1 into (2), we have:

2(x_2/3 - 1/3) + x_2 = 5/3 x_2 - 2/3 = 1

=> x_2 = 1
=> x_1 = 0

From (3), we have:

3x_3 - x_4 = 2

=> x_3 = 2/3 + x_4/3

From plugging x_3 into (4), we have:

2(2/3 + x_4/3) + x_4 = 5/3x_4 + 4/3 = 2

=> x_4 = 2/5
=> x_3 = 12/15 = 4/5

Therefore,

color(blue)(X = [(0,1),(4/5,2/5)])

[(0,1),(4/5,2/5)][(3a,2b),(-a,b)] = [(-a,b),(2a,2b)].