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प्रश्न
Find non singular matrices P and Q such that A = `[(1,2,3,2),(2,3,5,1),(1,3,4,5)]`
उत्तर
A = `[(1,2,3,2),(2,3,5,1),(1,3,4,5)]`
For PAQ form ,
`A=I_(3xx3).A_(3xx4).I_(3xx3)`
`[(1,2,3,2),(2,3,5,1),(1,3,4,5)]=[(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)]`
`R_2-2R_1, R_3-R_1,`
`[(1,2,3,2),(0,-1,-1,-3),(0,-1,1,3)]=[(1,0,0),(-2,1,0),(-1,0,1)]A[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]`
`C_2-2C_1, C_3-3C_1, C_4-2C_1,`
`[(1,0,0,0),(0,-1,-1,-3),(0,-1,1,3)]=[(1,0,0),(-2,1,0),(-1,0,1)]A[(1,-2,-3,-2),(0,1,0,0),(0,0,1,0),(0,0,0,1)]`
`R_3+R_2,`
`[(1,0,0,0),(0,-1,-1,-3),(0,0,0,0)]=[(1,0,0),(-2,1,0),(-3,1,1)]A[(1,-2,-3,-2),(0,1,0,0),(0,0,1,0),(0,0,0,1)]`
`C_3-C_2, C_4-3C_2,`
`[(1,0,0,0),(0,-1,0,0),(0,0,0,0)]=[(1,0,0),(-2,1,0),(-3,1,1)]A[(1,-2,-1,4),(0,1,-1,-3),(0,0,1,0),(0,0,0,1)]`
`-R_2,`
`[(1,0,0,0),(0,1,0,0),(0,0,0,0)]=[(1,0,0),(2,-1,0),(-3,1,1)]A[(1,-2,-1,4),(0,1,-1,-3),(0,0,1,0),(0,0,0,1)]`
Now A is in Normal form.
Compare this w.r.t A=PAQ form ,
`therefore "P"=[(1,0,0),(2,-1,0),(-3,1,1)]` `"Q"=[(1,-2,-1,4),(0,1,-1,-3),(0,0,1,0),(0,0,0,1)]`
∴ Rank of given matrix A is 2.
