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प्रश्न
Reduce matrix to PAQ normal form and find 2 non-Singular matrices P & Q.
`[[1,2,-1,2],[2,5,.2,3],[1,2,1,2]]`
उत्तर
`A_(3xx4)=I_(3xx3) xx A_(3xx4) xx I_(4xx4)`
`[[1,2,-1,2],[2,5,-2,3],[1,2,1,2]]=[[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,-1,2],[0,1,0,-1],[0,0,2,0]]=[[1,0,0],[-2,1,0],[-1,0,0]] A[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]`
`C_2-2C_1;C_3+C_1;C_4-2C_1;` → `[[1,0,0,0],[0,1,0,-1],[0,0,2,0]]=[[1,0,0],[-2,1,0],[-1,0,1]]A[[1,-2,1,-2],[0,1,0,0],[0,0,1,0],[0,0,0,1]]`
`C_4+C_2;1/2 C_3` → `[[1,0,0,0],[0,1,0,0],[0,0,1,0]]=[[1,0,0],[-2,1,0],[-1,0,1]] A [[1,-2,1/2,4],[0,1,0,1],[0,0,1/2,0],[0,0,0,1]]`
LHS is the required PAQ form.
Here, P = `[[1,0,0],[-2,1,0],[-1,0,1]] and Q=[[1,-2,1/2,-4],[0,1,0,1],[0,0,1/2,0],[0,0,0,1]]`
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