Quiz 1 Ma 322 005 Name ______________

1. A and B are 2 by 2 matrices whose inverses are [Maple Math] and [Maple Math] . Use the properties of inverses to calculate the inverse of AB.

We know [Maple Math] = [Maple Math] [Maple Math] = [Maple Math] [Maple Math] = [Maple Math]

2. A certain system of linear equations leads to the augmented matrix [Maple Math] , which when submitted to elimination leads to the matrix [Maple Math] .

a) Write down first elimination matrix [Maple Math] used to carry out the elimination.

[Maple Math] is the 3 by 3 which multiplies the 1st row by -3/2 and adds it to the 2nd row. [Maple Math]

b) Under what conditions (on a) does the system have a solution?

No pivot can occur in the augmented column. So in the last row, [Maple Math] , that is, a must be 3 in order that the system have a solution.

c) Assuming that the system has a solution, is the solution unique? Explain.

No. There are 2 equations in 4 variables. The last two variables can take any value. There is a whole plane of solutions.

3. I will tell you that [Maple Math] , and that the inverse of A is [Maple Math] .

a) What is t? Compute the 2,2 entry of the product which is the identity 3 by 3 matrix. It will be 1*2 + 2*t + 1*3 = 1. Solve for t to get t = - 2

b) Solve the matrix equation Ax = b where b = [Maple Math] . The easy way, since A^(-1) is known is

x = A^(-1) b = [Maple Math] = [Maple Math]