Let V M22 R be the set of 2 times 2 matrices with real entr

Let V = M_22 (R) be the set of 2 times 2 matrices with real entries. If A = (a b c d) is an element of V, then tr A = a + d is called the trace of A. a. Show that V is a vector space, and U = {A element V|tr A = 0} is a subspace (so-called, traceless matrices): b. Prove that S = {A element V|A^T = A} (symmetric matrices) and W = {A element V|A^T = -A} (skew-symmetric matrices) are subspaces of V: c. Prove that W is a subspace of U, but they do not coincide (U has more elements).

Solution

The trace of a matrix is defined as the sum of all the principal diagonal elements.

If a11=a;a12=b;a21=c;a22=d

Then trace of. Matrix is

a11+a22

=a+d