Essay on Matrix multiplication: basic

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DIAGNOSTIC IN-CLASS QUIZ: DUE FRIDAY OCTOBER 25: MATRIX
MULTIPLICATION (BASIC)
MATH 196, SECTION 57 (VIPUL NAIK)

Your name (print clearly in capital letters):
PLEASE DO NOT DISCUSS ANY QUESTIONS
This quiz tests for basic comprehension of the setup for matrix multiplication. It corresponds to the material from Sections 1-6 (excluding Section 4) of the Matrix multiplication and inversion notes, and also to Section 2.3 of the book.
(1) Do not discuss this!: Suppose A and B are (not necessarily square) matrices. Then, which of the following describes correctly the relationship between the existence and value of the (alleged) matrix product AB and the existence and value of the (alleged) matrix product BA?
(A) AB is defined if and only if BA is defined, and if so, they are equal.
(B) AB is defined if and only if BA is defined, but they need not be equal.
(C) If AB and BA are both defined, then AB = BA. However, it is possible for one of AB and BA to be defined and the other to not be defined.
(D) It is possible for only one of AB and BA to be defined. It is also possible for both AB and BA to be defined, but to not be equal to each other.
Your answer:
(2) Do not discuss this!: Suppose A and B are matrices such that both AB and BA are defined. Which of the following correctly describes what we know about AB and BA?
(A) Both AB and BA are square matrices and have the same dimensions, i.e., in both AB and
BA, the number of rows equals the number of columns, and further, the number of rows of AB equals the number of rows of BA.
(B) Both AB and BA are square matrices (the number of rows equals the number of columns) but they may not have the same dimensions: the number of rows in AB need not equal the number of rows in BA.
(C) AB and BA need not be square matrices but both must have the same dimensions: the number of rows in AB equals the number of rows in BA, and the number of columns in AB equals the number of columns in BA.
(D) AB and BA need not be square matrices and they need not have the same row count or the same column count, i.e., the number of rows in AB need not equal the number of rows in BA, and the number of columns in AB need not equal the number of columns in BA.
Your answer:
(3) Do not discuss this!: Suppose A and B are matrices such that both AB and A + B are defined.
Which of the following correctly describes what we know about A and B?
(A) Both A and B are square matrices and have the same dimensions, i.e., in both A and B, the number of rows equals the number of columns, and further, the number of rows of A equals the number of rows of B.
(B) Both A and B are square matrices (the number of rows equals the number of columns) but they may not have the same dimensions: the number of rows in A need not equal the number of rows in B.
(C) A and B need not be square matrices but both must have the same dimensions: the number of rows in A equals the number of rows in B, and the number of columns in A equals the number of columns in B.
1

(D) A and B need not be square matrices and they need not have the same row count or the same column count, i.e., the number of rows in A need not equal the number of rows in B, and the number of columns in A need not equal the number of columns in B.
Your answer:
(4) Do not discuss this!: Suppose A is a p × q matrix and B is a q × r matrix. The product matrix
AB is a p × r matrix. Using the convention of matrices as linear transformations via their action by multiplication on column vectors, what is the appropriate interpretation of the matrix product in terms of composing linear transformations?
(A) A corresponds
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