Temple University
Department of Economics
Econ 8009
Fall 2011

Linear Algebra Hwk 
Question 1.

First we'll add B+C, then multiply from the left by A:

Now we'll construct AB + AC:

By inspection you can see we got he same answer both ways. The simple example I used is scalable up to as many rows and columns as you want as long as things are conformable.

Question 2
By "makes sense" it must be meant that A and B are conformable for multiplication such as A*B; since matrix multiplication is not commutative, the reverse order is not necessarily true. For this specific question it is said that (AB)C makes sense. Therefore, if A: nxk and B:kxm then AB: nxm. It is also said that (AB)C makes sense, so C must be mxr.  so BC also makes sense, and therefore A(BC) does so as well. Now consider an example using A, B, and C defined as

Now do the multiplication the second way

By inspection you can see that the two results would be the same if we carried out the final step in each.

Question 3. We'll do this one with a simple case. Consider a matrix A: 2x2.

The stipulation A*A = A means that a12*a21 = 0 and that a11 = a112.  Since A11 is the square of a number it must be positive. The same goes for a22. Therefore the trace is positive.
Question 4

(a) The inverse:

(b) The eigenvalues, their inverses, their product, and their sum.

(c) The determinant and trace of A

(d) The sum of the eigen values is equal to the trace of A.
(e) The product of the eigen values is equal to the determinant of A
(f) The eigenvalues, the determinant and trace of A-1.

(g)  The eigenvalues of A and the eigenvalues of A-1 are reciprocals of one another.

The determinant of A is the reciprocal of the determinant of A-1.