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Final Practice Exam

Math 3A: Summer II 2024

Problem 1. (15 points) Given 

(a) Determine the eigenvalues and corresponding eigenvetors of A.

(b) Diagonalize A.

Problem 2. (15 points) Given 

(a) Find a basis for the nullspace of A.

(b) Verify that the Rank-Nullity Theorem applies to A.

Problem 3. (10 points) Let

and

Write down the matrices that take  to 

Problem 4. (20 points) Determine if the following statements are true or false. Explain why.

(a) If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent.

(b) If a matrix is in reduced row echelon form, then it is also in row echelon form.

(c) Let A, B, C be matrices. If AC = BC and C ≠ 0, then A = B.

(d) If B has a column of zeros, then so does AB if this product is defined.

(e) Let A, B be square matrices of the same size. Then A2 − B2 = (A + B)(A − B).

Problem 5. (15 points) Suppose that  is a basis for the nullspace of the matrix A − 3I3 and that  is a basis for the nullspace of the matrix A + 5I3.

(a) Write  as a linear combination of  and 

(b) Find 

Problem 6. (10 points) Let

Find a diagonal matrix D and an invertible matrix P such that A = PDP −1. Find A10.

Problem 7. (15 points) Given the matrix  find the characteristic polynomial of H and determine if H is diagonalizable. If it is, find the diagonal matrix and the corresponding eigenbasis.