1.

Evaluate the determinants in(i) `|(costheta,-sintheta),(sintheta,costheta)|` (ii) `|(x^2-x+1,x-1),(x+1,x+1)|`

Answer» (i) `|[costheta,-sintheta],[sintheta,costheta]|`
`=[cos^2theta - (-sin^2theta)]`
`=[cos^2theta + sin^2theta)]`
`=1`
`:. |[costheta,-sintheta],[sintheta,costheta]| = 1`
(ii) `|[x^2-x+1,x-1],[x+1,x+1]|`
`=[(x+1)(x^2-x+1)-(x+1)(x-1)]`
`=[(x+1)(x^2-x+1-x+1)]`
`=[(x+1)(x^2-2x+2)]`
`=x^3-2x^2+2x+x^2-2x+2`
`=x^3-x^2+2`
`:.|[x^2-x+1,x-1],[x+1,x+1]| =x^3-x^2+2`
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