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Let A be a `2 xx 2` matrix with non-zero entries and let A^2=I, where i is a `2 xx 2` identity matrix, Tr(A) i= sum of diagonal elements of A and `|A|` = determinant of matrix A.Statement 1:Tr(A)=0Statement 2:`|A|`=1 |
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Answer» Let `A = [[a,b],[c,d]]` It is given that, `A^2 = I` `:. [[a,b],[c,d]][[a,b],[c,d]] = [[1,0],[0,1]]` `=>[[a^2+bc,ab+bd],[ac+cd,bc+d^2]]= [[1,0],[0,1]]` `=>a^2+bc = 1->(1)` `=>ab+bd = 0 =>b(a+d) = 0 => a = -d->(2) ...[As b!=0]` So, we can write, `A = [[a,b],[c,-a]]` `:. Tr(A) = a+(-a) = 0` `|A| = -a^2-bc =-(a^2+bc) = -1` So, first statement is true but second statement is false. |
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