Math question Let A be a matrix such that A^2 = I where I is the identity matrix, Show that e^At = cosh(t)I + sinh(t) A. (Hint: use identities cos h(t) = cos(it) and sinh(t) = -isin(it) and the power series expansion of cosh(t) and sinh(t)). Solution cosht = cos (it) sinht = -isinit Hence coshtI + sinhtA = cos(it)I -isin(it) A = cos it (A2) - i sin (it) A = A(cosit-sin itA) = eAt.

1. Math question Let A be a matrix such that A^2 = I where I is the identity matrix, Show that e^At
= cosh(t)I + sinh(t) A. (Hint: use identities cos h(t) = cos(it) and sinh(t) = -isin(it) and the power
series expansion of cosh(t) and sinh(t)).
Solution
cosht = cos (it)
sinht = -isinit
Hence coshtI + sinhtA
= cos(it)I -isin(it) A
= cos it (A2) - i sin (it) A
= A(cosit-sin itA)
= eAt