Midterm II Fall 2025 MATH 220 Matrices After we finish the material on 3 27 try to take this like a real exam so you can get a feel for the timing and spot any topics you still need to review NOTE This practice is longer than the actual exam so you should allow yourself 90 minutes rather than the standard 75 minutes PRINT your name CLEARLY as it appears in CANVAS and LionPath psu edu Name PSU Email ID Section Instructor Section number INSTRUCTIONS Write your initials on the bottom of each page in the indicated space Failure to do this may result in a 5 point deduction Check that your exam contains 14 questions Answer all questions in your test booklet show all of your work write legibly and write your final answer in the box or space provided The use of a calculator cell phone smartwatch or any other electronic device is not permitted during this examination The use of scrap paper or notes of any kind is not permitted Good luck Page 1 of 14 Initials You can use this page for overflow scratch work You must label which problem s it pertains to and indicate on the page of the problem that it can be found on page 2 Only use this page if you run out of space elsewhere Page 2 of 14 Initials 1 7 points Let A B and C cid 20 1 2 1 cid 21 3 0 2 cid 20 0 1 1 cid 21 2 1 0 2 1 1 2 0 3 For each of the following If the expression is undefined fill in the bubble and don t write perform the computation anything in the box a C T 3B b AB c CB d A2 C T 3B The expression is undefined The expression is undefined The expression is undefined AB CB A2 The expression is undefined Page 3 of 14 Initials 2 8 points For each of the following matrices calculate the determinant then determine if each is invertible or singular then determine whether the matrix is invertible 0 2 1 0 0 3 4 4 4 0 4 2 3 0 3 0 a A 0 2 3 0 0 1 0 1 2 2 3 0 0 1 1 0 b B Determinant A is invertible A is not invertible Determinant B is invertible B is not invertible Page 4 of 14 Initials 3 3 points Let A B and C be n n invertible matrices Simplify the expression completely AC 1 1 BAT T C 1 1 0 2 h 1 4 1 1 2 Compute the determinant of A then find all values of h 4 5 points Let A such that the matrix A is invertible Determinant Value s of h Page 5 of 14 Initials cid 20 1 2 cid 21 3 8 5 4 points Let A Use the matrix inverse A 1 to solve the linear system You must show work below using the given inverse to find your solution to receive any credit cid 40 x 2y 2 3x 8y 4 cid 20 4 1 3 2 1 2 cid 21 Solution 6 6 points Cells in a Petri dish can have two phases active or lazy Each day 4 of the active cells remain active and the other 1 3 All lazy cells become active 4 become lazy The number of cells after one day can be predicted using the transition matrix cid 20 3 cid 21 4 1 1 4 0 a Give the transition matrix which predicts the number of cells in each phase after two days Complete all computations b Give the transition matrix which predicts the number of cells in each phase from one day ago Complete all computations Transition Matrix Transition Matrix Page 6 of 14 Initials 7 7 points Let A 1 1 2 1 2 4 5 2 3 2 4 5 8 Find A 1 if it exists or show that A does not have an inverse If A does not have an inverse fill in the bubble and don t write anything in the box A does not have an inverse A 1 Page 7 of 14 Initials 8 9 points Select the best answer from the given choices for each of the following questions a If T R8 R3 is a linear transformation and T is onto then the nullity of T is 3 True False True False b If A is a 3 3 matrix with eigenvalues 0 1 and 3 then A must be invertible c If T R3 R3 is a linear transformation which is one to one then the images of the standard basis vectors also form a basis of R3 True False d Is the null space of a transformation T a subspace of the domain or codomain of T Domain Codomain e T R3 R3 is a linear transformation whose standard matrix is B whose columns of B are linearly dependent Which of the following must be true about the determinant of B det B 0 det B 0 det B 0 Not enough information f Suppose that A is an n n matrix with det A 3 Select all of the following statements 1 3 that must be true det A 1 det 3A 9 det A3 27 det AT 1 3 Page 8 of 14 Initials 9 8 points Determine if each subset is a subspace Provide a complete justify for your answer cid 26 cid 20 1 cid 21 x cid 12 cid 12 cid 12 cid 12 x R cid 27 a Is S R2 a subspace of R2 This set is a subspace of R2 This set is not a subspace of R2 Justification b Is S a subspace of R2 cid 21 cid 26 cid 20 x x R2 cid 12 cid 12 cid 12 cid 12 x R cid 27 This set is a subspace of R2 This set is not a subspace of R2 Justification Page 9 of 14 Initials 10 8 points For each of the following sets of vectors determine if the set is a basis for the corresponding vector space then explain why using at least one complete sentence cid 26 cid 20 3 cid 21 cid 20 2 cid 21 cid 27 1 7 a for R2 Is a basis Not a basis Explanation 3 0 0 1 1 2 2 0 0 for R3 b Is a basis Not a basis Explanation 1 0 3 0 0 1 1 0 2 2 0 5 for R4 c Is a basis Not a basis Explanation Page 10 of 14 Initials 11 11 points Let T be a linear transformation whose standard matrix A is given below The reduced echelon form of A is given as well 2 0 4 2 8 1 1 3 9 0 3 3 15 1 0 2 1 4 3 4 Gauss Jordan 1 0 2 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 3 0 A a What are the rank and nullity of …