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Prove, using properties of determinants:...

Prove, using properties of determinants: `|[y+k, y, y],[ y, y+k, y],[ y, y, y+k]|=k^2(3y+k)`

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`L.H.S. = |[y+k,y,y],[y,y+k,y],[y,y,y+k]|`
Applying `R_1->R_1+R_2+R_3`
`=|[3y+k,3y+k,3y+k],[y,y+k,y],[y,y,y+k]|`
Applying `C_2->C_2-C_1 and C_3->C_3-C_1`
`=|[3y+k,0,0],[y,k,0],[y,0,k]|`
`=(3y+k)(k^2-0)-0+0 `
`=k^2(3y+k) = R.H.S.`
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