Question

Find an example of 2 by2 matrices with a12 = .5 for which:

a. A^2 = I
b. A^-1 = A^T
c. A^2 = A

By the way, a12 means the first row and second column.

Answer 1

I will do the first one:
let A =
a .5
b c

the A^2 =
(a^2+b/2) (a/2+c/2)
(ab + bc) (b/2+c^2)

then
a^2+b/2=1 ---> 2a^2 + b = 2 (#1)
a/2+c/2=0 ---> a+c=0 or a = -c (#2)
ab+bc=0 -----> b(a+c) = 0 (#3)
b/2+c^2=1 ---> b+2c^2=2 (#4)

from #3
b(a+c)=0 , but a+c=0
so b = 0
in #1
2a^2 + 0 = 2
a^2 = 1
a = 1
then c=-1

so A =
1 .5
0 -1

for the others, you should have learned a quick way to find the inverse of a 2by2
T stands for transpose,
to take the transpose, your rows of the first matrix become the columns of the the transpose.