Quiz 2 Ma 322 005 Name ______________

1. Let [Maple Math] be the 5 by 5 identity matrix with its ith and jth row interchanged. So for example, [Maple Math] .

a) The product matrix [Maple Math] is just the identity matrix. What is a simple explanation for this?

We know [Maple Math] A is A with its its ith and jth rows interchanged. When A = [Maple Math] , we are back to the identity matrix.

b) Let P be the product matrix [Maple Math] . Compute P

[Maple Math] = [Maple Math] = [Maple Math] = [Maple Math] = [Maple Math]

c) The determinant of P is -1. Why is that?

There are an odd number of row interchanges. Follows from properties 1 and 2 of determinants.

2. Let [Maple Math] .

a) Find the LU factorization of A and use it to compute the determinant of A.

Take [Maple Math] Then [Maple Math] = [Maple Math] = U and L = [Maple Math] = [Maple Math]

b) Use Gauss-Jordan elimination to find the inverse of A.

Multiply [Maple Math] by [Maple Math] and then by D = [Maple Math] to get [Maple Math] = I so [Maple Math] = [Maple Math]

3. a) Express the angle made by the vector [1,2,3] and a vector [x,y,0] in the xy-plane in terms of x and y.

angle between u and v is [Maple Math] = arccos( [Maple Math] )

b) Geometrically, the above angle should be smallest when (x,y) is a positive scalar multiple of (1,2). Use your calcluator to calculate the angle in degrees.

arccos( [Maple Math] ) [Maple Math] = [Maple Math] degrees