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Question
Find the non-singular matrices P & Q such that PAQ is in normal form where`[(1,2,3,4),(2,1,4,3),(3,0,5,-10)]`
Solution
Given Matrix is A =`[(1,2,3,4),(2,1,4,3),(3,0,5,-10)]`
The order of matrix is 3 X 4
∴ A=I3 A I4.
`[(1,2,3,4),(2,1,4,3),(3,0,5,-10)]=[(1,0,0),(0,1,0),(0,0,1)]"A"[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]`
Operate R2-2R1; R3-3R1
`[(1,2,3,4),(0,-3,-2,-5),(0,-6,-4,-22)]=[(1,0,0),(2,1,0),(3,0,1)]"A"[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]`
Operate C2-2C1; C3 -3C1; C4-4C1
`[(1,0,0,0),(0,-3,-2,-5),(0,-6,-4,-22)]=[(1,0,0),(2,1,0),(3,0,1)]"A"[(1,-2,-3,-4),(0,1,0,0),(0,0,1,0),(0,0,0,1)]`
Operate`c_2/(-3);c_3/(-2)`
`[(1,0,0,0),(0,1,1,-5),(0,-2,-2,-22)]=[(1,0,0),(-2,1,0),(-3,0,1)]"A"[(1,2/3,3/2,-4),(0,(-1)/3,0,0),(0,0,(-1)/2,0),(0,0,0,1)]`
Operate R3-2R2
`[(1,0,0,0),(0,1,1,-5),(0,0,0,-12)]=[(1,0,0),(-2,1,0),(1,-2,1)]"A"[(1,2/3,3/2,-4),(0,(-1)/3,0,0),(0,0,(-1)/2,0),(0,0,0,1)]`
Operate C34
`[(1,0,0,0),(0,1,1,0),(0,0,-12,0)]=[(1,0,0),(-2,1,0),(1,-2,1)]"A"[(1,2/3,(-2)/3,5/6),(0,(-1)/3,(-5)/3,1/3),(0,0,0,(-1)/2),(0,0,1,0)]`
Operate `R_3/(-12)`
`[(1,0,0,0),(0,1,0,0),(0,0,1,0)]=[(1,0,0),(-2,1,0),((-1)/12,1/6,(-1)/12)]"A"[(1,2/3,(-2)/3,5/6),(0,(-1)/3,(-5)/3,1/3),(0,0,0,(-1)/2),(0,0,1,0)]`
[I3, 0] = PAQ ie PAQ is in normal form,
Where, P = `[(1,0,0),(-2,1,0),((-1)/12,1/6,(-1)/12)]` and Q =`[(1,2/3,(-2)/3,5/6),(0,(-1)/3,(-5)/3,1/3),(0,0,0,(-1)/2),(0,0,1,0)]`
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