Math 410 – Linear Algebra Name
Exam I A – September 26, 2017
This is a closed book exam. No notes, or study aides are allowed. Calculators
that are not capable of symbolic manipulation may be used. Calculators that
are capable of symbolic manipulation (Ti-89, Ti-92, NSpire, etc) are not
permitted. Unless otherwise specified, each problem is worth 12 points. Partial
credit will be given for legible work. No credit will be given for unjustified
answers. Good Luck.
1. Determine whether the matrix is invertible. If it is, find its inverse. If it is not, explain
your reasoning.
2. Suppose is a 2 x 3 matrix and is a vector in . Prove that if
then .
3. Use the following matrices to compute the indicated expression, if it is defined.
4. Evaluate the determinant by first reducing the matrix to row echelon form.
5. Determine whether the statement is true or false, and justify your answer.
If and are square matrices and is invertible, then .
If a linear system has more unknowns than equations, then it must have infinitely
many solutions.
is a symmetric matrix if and only if is a symmetric matrix.
6. Complete the following theorem.
Theorem 2.3.8 - If is an matrix, then the following statements are
equivalent.
(a) is invertible.
7. Let . Find the standard matrix for the linear transformation
defined by .
8. Find each of the following with and .
The parametric equation of the line through and parallel to .
The angle between the vectors and .
The equation of the plane through the origin parallel to both and .