Find the matrix X so that `X[1 2 3 4 5 6]=[-7-8-9""""""2""""""4""""6]`

Answer

`X=[(1,-2),(2,0)]`

Explanation

It is given that
`X[(1,2,3),(4,5,6)]=[(-7,-8,-9),(2,4,6)]`
The matrix given on the R.H.S. of the equation is a `2xx3` matrix and the one given on the L.H.S is a `2xx3` matrix. Therefore, X has to be a `2xx2` matrix.
Now, let `X=[(a,c),(b,d)]`
therefore, we have
`[(a,c),(b,d)][(1,2,3),(4,5,6)]=[(-7,-8,-9),(2,4,6)]`
or `[(a+4c,2a+5c,3a+6c),(b+4d,2b+5d,3b+6d)]=[(-7,-8,-9),(2,4,6)]`
Equating the corresponding elements of the two matrices, we have
`{:(a+4c=-7",",2a+5c=-8",",3a+6c=-9),(b+4d=2",",2b+5d=4",",3b+6d=6):}`
Now, `a+4c=-7 implies a=-7-4c`
`:. 2a+5c=-8 implies -14-8c+5c=-8`
or `-3c=6`
or `c=-2`
`:. a=-7-4(-2)=-7+8=1`
Now, `b+4d=2` or `b=2-4d`
`:. 2b+2d=4implies 4-8d+5d=4`
`implies -3d=0`
`d=0`
`:. b=2-4(0)=2`
Thus, `a=1, b=2, c=-2, d=0`
Hence, the required matrix X is `[(1,-2),(2,0)]`