By using properties of determinants. Show that: (i) `|(x+4,2x,2x),(2x,x+4,2x),(2x,2x,x+4)|=(5x-4)(4-x)^2` (ii) `|(y+k,y,y),(y,y+k,y),(y,y,y+k)|=k^2(3y+k)`

Explanation

(i) `L.H.S. = |[x+4,2x,2x],[2x,x+4,2x],[2x,2x,x+4]|`

Applying `C_1->C_1+C_2+C_3`

`= |[5x+4,2x,2x],[5x+4,x+4,2x],[5x+4,2x,x+4]|`

`=(5x+4)|[1,2x,2x],[1,x+4,2x],[1,2x,x+4]|`

Applying `R_2->R_2-R_1` and `R_3->R_3-R_1`

`=(5x+4)|[1,2x,2x],[0,4-x,0],[0,0,4-x]|`

`=(5x+4)(4-x)^2|[1,2x,2x],[0,1,0],[0,0,1]|`

`=(5x+4)(4-x)^2[1(1-0)-0+0]`

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