Matrix question? (Might be a little complicated)

Question

Right. So I'm given these 3 matrices:

A= { 3 5 -2 }             B=  { 8    -5   -4    -1 }               C= { 4 3  2 }

      { -4 6 7 }                   { 49  32   61   40 }                   { 7 0 -3 }

      { 1  2 0 }                    { 7    -1    4      4 }                     { 8 6  4  }

 

I have to find Matrix X so that AX=B. and Show the matrix set-up, or explain why it's impossible, if  it is.

Answer 1

Left multiply by A^-1 (A inverse) to get X = A^-1 * B.  A-inverse exists so long as the determinant of A does not equal zero.

Comments

Thank you, I did that and found X... but when i checked the answer (By multiplying X by A) it said ERROR and that the dimensions weren't the same, so does that mean it's impossible?

Answer 2

AX=B so if A' is the inverse of A: A'AX=A'B and X=A'B.

The determinant of A or |A|=(0+35+16)-(-12+42+0)=51-30=21 which is non-zero, so A' can be found.

Compute the inverse:

( -14 -7 -14 )......( -14     7 -14 ).....( -14 -4  47 )............( -14 -4  47 )

(    4  2     1 ) ⇒ (   -4     2   -1 ) ⇒ (    7  2 -29 ) ⇒ 1/21(    7  2 -29 )

(  47  29 38 ).....(   47 -29   38 ).....( -14 -1  38 )............( -14 -1  38 )
 

Compute X=A'B:

.......( -14 -4  47 ) (  8   -5 -4  -1 )......(  1       -5          0            2           )

1/21(    7  2 -29 ) ( 49 32 61 40 ) ⇒ (  433/3 638/21 1858/21 1417/21 )

.......( -14 -1  38 ) (  7  -1   4   4 )......( -445/3 760/21     47/21 -394/21 )