Assume A = [[5,-2],[-6,-7]] and b = [[1],[2]]

Write b as a linear combination of the columns of A.

b = __ [[5],[-2]] + ____ [[-6],[-7]]

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`A=[[5,-2],[-6,-7]]` `b=[[1],[2]]`

called x and y coefficients of combination we have the system:

`5x-2y= 1`

`-6x-7y=2`

A solution exists if the detrminant A `!=0`

Indeed `det A =-47`

So we have to calculate the determinats:

`det[[1,-2],[2,-7]]=-3`

and:

`det [[5,1],[-6,2]]` `=16`

so that:

`x=(det[[1,-2],[2,-7]])/(det[[5,-2],[-6,-7]])` `=3/47`

`y=(det[[5,1],[-6,2]])/(det[[5,-2],[-6,-7]])` `=-16/47`

Let

`b=lambda[[5],[-2]]+mu[[-6],[-7]]` , and `lambda,mu!=0`

`[[1],[2]]=lambda[[5],[-2]]+mu[[-6],[-7]]`

`1=5lambda-6mu`

`2=-2lambda-7mu`

Solving above system of equations for `lambda ,and mu .` We have `mu=-12/47,lambda=-5/47`

`b=(-5/47)[[5],[-2]]+(-12/47)[[-6],[-7]]`

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