As a remainder of some definitions weve seen A matrix A is c

As a remainder of some definitions we\'ve seen: A matrix A is called involutory if A^2 = I. A matrix A is called orthogonal if A(A^T) = I. A matrix is called symmetric if A = A^T. Suppose A is an involutory matrix. Show that under this position, A is orthogonal if and only if A is symmetric. Give an example of a 2 Times 2 matrix A satisfying all properties that is NOT diagonal.

Solution

Solution: Given that A is an involutory matrix. So A²= I.

Suppose that A is orthogonal. we have to show that A is symmetric.

Since A is involutory and orthoganal, so A²= I and AA^T = I.

Implies that A²= I =AA^T

Implies that AA= I =AA^T

Implies that A^-1(AA) =A^-1(AA^T)

Implies that (A^-1A)A =(A^-1A)A^T (by associativity)

Implies that IA =IA^T

Implies that A =A^T

Therefore A is symmetric.

Conversely, suppose that A is symmetric. We have to show that A is orthoganal.

Since A is symmetric, so A =A^T

Since A is an involutory matrix, A²= I.

Implies that I = A²= A.A =A.A^T( As A =A^T)

So, A.A^T =I

Hence A is orthogonal.

Example: Let A = (a_ij)_2x2=

Then A.A = I , A.A^T =I, A=A^T

Hence A is involutory, symmetric and orthogonal.

0	1
1	0