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If A is a aquare matrix of order 2 and d...

If A is a aquare matrix of order 2 and det. `A=10`, then `((tr. A)^(2)-tr. (A^(2)))` is equal to ______ .

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The correct Answer is:
20
Let `A=[(a,b),(c,d)]`
`:. A^(2)=[(a,b),(c,d)][(a,b),(c,d)]`
`=[(a^(2)+bc,ab+bd),(ac+cd,bc+d^(2))]`
`:. ("tr. A")^(2)-("tr. "A^(2))=(a+d)^(2)-(a^(2)+bc+bc+d^(2))`
`=2(ad-bc)=2` det. `A=20`
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