Correct Answer - Option 1 : 2
Concept:
Let A = [aij]n×n be square matrix. Then the sum of elements lying along the principal diagonal is said to be trace of A and write as,
Trace of A tr A = a11 + a22 + --- ann ---(1)
Calculation:-
Given,
\(A = \left[ {\begin{array}{*{20}{c}} {\cosh x}&{\sinh x}\\ { - \sinh x}&{\cosh x} \end{array}} \right]\)
\({A^2} = A \times A\)
\( = \left[ {\begin{array}{*{20}{c}} {\cosh x}&{\sinh x}\\ { - \sinh x}&{\cosh x} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\cosh x}&{\sinh x}\\ { - \sinh x}&{\cosh x} \end{array}} \right]\)
\( = \left[ {\begin{array}{*{20}{c}} {{{\cosh }^2}x - {{\sinh }^2}x}&{\cosh x\sinh x + \sinh x\cosh x}\\ { - \sinh x\cosh x - \sinh x\cosh x}&{ - {{\sinh }^2}x + {{\cosh }^2}x} \end{array}} \right]\) ---(A)
As, we know, , then (A) becomes
\({A^2} = \left[ {\begin{array}{*{20}{c}} {{{\cosh }^2}x - {{\sinh }^2}x}&{2\sinh x\cosh x}\\ { - 2\sinh x\cosh x}&{{{\cosh }^2}x - {{\sinh }^2}x} \end{array}} \right]\)
\({A^2} = \left[ {\begin{array}{*{20}{c}} 1&{2\sinh x\cosh x}\\ { - 2\sinh x\cosh x}&1 \end{array}} \right]\)
From equation (1) tr (A2) = 1 + 1 = 2
