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.
Above is a single matrix A shown.
Above shown is a single matrix B.
1 Answer
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
#3x_1 - x_2 = -1#
#=> x_1 = x_2/3 - 1/3#
From plugging
#2(x_2/3 - 1/3) + x_2 = 5/3 x_2 - 2/3 = 1#
#=> x_2 = 1#
#=> x_1 = 0#
From
#3x_3 - x_4 = 2#
#=> x_3 = 2/3 + x_4/3#
From plugging
#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)]# .