Linear Algebra - Fall 2023
Exam 1
1. (a)
Determine if A is invertible, and if so determine its inverse.
(b) Suppose B, C are 3⇥3 invertible matrices such that BC = A. Calculate the product C−1B−1.
2. The row reduced echelon form. of the augmented matrix of a linear system is
State whether the system is consistent or inconsistent and why; find the solution of the system if it has solutions, and state which variables are leading variables and which are free variables.
3. Four matrices are given below, each being the row reduced echelon form. of a matrix of a system of linear equations. For each of these, determine the number of solutions of the original system; briefly justify:
0 (zero, or inconsistent system), 1 (unique) or 1 (infinitely many solutions).
(a)
(b)
(c)
(d)
4. Write the matrix of the linear transformation that represents each of the following:
(a) A rotation about the origin of angle 2/π counterclockwise (you MUST simplify your answer).
(b) A reflection about the line x1 = x2.
(c) The transformation that is obtained by doing a rotation about the origin of angle 2/π counter-clockwise FIRST, followed by a reflection about the line x1 = x2. (that is, doing (a) first, then (b)).
5. (a) Let Calculate det(A), determine if A is an invertible matrix and find its inverse if it is.
(b) For what values of k is the matrix
invertible?
6. A middle school student was trying to solve a linear system. The pencil written notebook took some water damage, and some writing was no longer visible. The system was of the form.
where a and b represent numbers that are no longer visible. Below the system, the student wrote several solutions to the system (but which now can no longer be clearly distinguished). Reconstruct this part of the notebook and find the numbers a and b. JUSTIFY your answer.
7. Consider the matrix
(a) Interpret the linear transformation T(x) = Ax geometrically. (Write a sentence that explains its effect, and/or draw a picture.)
(b) Possibly using (a), calculate A20.