1. Prove the following formulas are satisfied by the trace matrix operator. a. trace (A + B) = trace (A) + trace (B) b. trace (cA) = c trace (A), where c is a scalar. 2. For an arbitrary  column...

1. Prove the following formulas are satisfied by the trace matrix operator.

a. trace (A + B) = trace (A) + trace (B)

b. trace (cA) = c trace (A), where c is a scalar.

2. For an arbitrary
 column vector and an
 matrix
 show that

^T
 is a real number. This is called a quadratic form. For

^T
 compute

^T
 for the matrix

3. Prove the following properties of the matrix transpose operator.

4. Prove that

<sup>T

T

T</sup>
if is
 and
 is
 Use the definition of matrix multiplication and the fact that taking the transpose means element

_ij
of A is the element at row
 column
 of

^T