Linear algebra and differential equations problem

Qno.2
a. A and B will have same size
b. Trace A = Trace B
c. A^T need not be equal to B^T
d. Yes, rank A = rank B
e. True
If A is symmetric B is also symmetric otherwise not.
Qno.3
Since A can be expressed as a product of elementary matrices we have that A is invertible. Since A is invertible A inverse exists
Hence X = A_1B is the unique solution for X
Qno.4
a. A =
So
Thus square the elements of diagonal and writte in corresponding diagonal places.
b. In the same lines as above we can see
For nxn matrix diagonal elements in A^n would be square of corresponding diagonal elements in A
c. Let us assume A^k=
Already true for n=1
Hence A^n = for all natural numbers
d. Extending the...