Did you read my tips on how to study and learn this course?  If not, here is a link to those important suggestions:

Did you miss my Tips for Assignment 1? Click here.

Here is a link to the actual assignment, in case you don't have it handy:

Study Lesson 3 (Matrix Math), Lesson 4 (The Inverse of a Matrix and Applications), Lesson 5 (Elementary Matrices), and Lesson 6 (Determinants and Their Properties) from my Linear Algebra & Vector Geometry book to prepare for this assignment.

Do note that, although not included on this assignment, Lesson 7 (The Adjoint Matrix) and Lesson 8 (Cramer's Rule) are on the upcoming midterm exam.  The midterm exam covers Lessons 1 to 8 in my book.

This is just classic matrix math stuff as taught in Lesson 3 of my book.

The trace of a matrix is just the sum of the numbers on the main diagonal of a square matrix. First, the matrix must be square, n by n. Then trace(A) is just 1,1 + 2,2 + 3,3 + etc.

Classic inverse matrix stuff as taught in Lesson 4 of my book.  Note that you can verify that your answer is correct by confirming that the product of A and A-inverse is the Identity matrix.

Read the Equivalent Statements Theorem in Lesson 6, page 189 of my book. A determinant is the best approach here.  Then, recall what I say about solutions to homogeneous systems of equations back in Lesson 2, page 49 of my book.

This is just classic elementary matrix stuff as taught in Lesson 5 of my book.  Question 5 is especially similar.

Standard determinant problem similar to my Lesson 6, question 1.  Note that you can check your answers by using a different method or a different row or column to expand along.

Part (e) is using the method I teach in Lesson 6, question 6.  Note that you can also still check your answer by using the expansion method. And, you can check the earlier answers by using the row-reduction method if you want.