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Find non singular matrices P & Q such that PAQ is in normal form where A ⎡ ⎢ ⎣ 2 − 2 3 3 − 1 2 1 2 − 1 ⎤ ⎥ ⎦ - Applied Mathematics 1

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Find non singular matrices P & Q such that PAQ is in normal form where A `[[2,-2,3],[3,-1,2],[1,2,-1]]`

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Solution

Matrix in PAQ form is given by , 

A=P A Q 

`[[2,-2,3],[3,-1,2],[1,2,-1]]`=`[[1,0,0],[0,1,0],[0,0,1]] A[[1,0,0],[0,1,0],[0,0,1]] `

`R_1→R_3` ,

`[[1,2,-1],[3,-1,2],[2,-2,-3]] =[[1,0,0],[0,1,0],[0,0,1]] A[[1,0,0],[0,1,0],[0,0,1]] `

`R_2-3R_1,R_3-2R_1`

`[[1,2,-1],[0,-7,5],[0,-6,5]]=[[0,0,1],[0,1,-3],[1,0,-2]] A[[1,0,0],[0,1,0],[0,0,1]] `

𝑪𝟐−𝟐𝑪𝟏,𝑪𝟑+𝑪𝟏,

Now A is in normal form with rank 3.
Compare with PAQ form ,

Concept: Reduction to Normal Form
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