If A is a 3×4 matrix and B is a 4×2 matrix, what is the size of (AB)^T?
A3×2
B2×3
C4×3
D2×4
AB is a 3×2 matrix (3 rows of A times 2 columns of B). Transposing swaps rows and columns, so (AB)^T is 2×3. Confirming via the rule: (AB)^T = B^T A^T, where B^T is 2×4 and A^T is 4×3, giving a 2×3 product. The order-reversal in the product rule is essential — B^T A^T is defined and gives 2×3, while A^T B^T would require multiplying 4×3 by 2×4, which is not defined.
Question 2 Multiple Choice
Which of the following statements about the transpose operation is FALSE?
A(A^T)^T = A
B(AB)^T = A^T B^T
C(A + B)^T = A^T + B^T
DIf A = A^T, then A is called a symmetric matrix
Option B is false: (AB)^T = B^T A^T, not A^T B^T. The order must be reversed — this is one of the most important properties of the transpose and is analogous to the inverse rule (AB)^{-1} = B^{-1}A^{-1}. Options A, C, and D are all true: transposing twice returns the original, the transpose distributes over addition without reversing order, and A = A^T is precisely the definition of a symmetric matrix.
Question 3 True / False
A symmetric matrix can be non-square (e.g., a 3×2 matrix can satisfy A = A^T).
TTrue
FFalse
Answer: False
A = A^T requires the matrix and its transpose to have the same dimensions. If A is m×n, then A^T is n×m. For A = A^T to hold, we need m = n — A must be square. Beyond being square, entry (i,j) must equal entry (j,i) for all i and j, meaning the matrix is symmetric across its main diagonal. Non-square matrices cannot satisfy A = A^T.
Question 4 True / False
For column vectors u and v of the same length, u^T v equals the dot product u · v.
TTrue
FFalse
Answer: True
If u and v are n×1 column vectors, then u^T is a 1×n row vector, and u^T v is a 1×n matrix times an n×1 matrix — the result is a 1×1 scalar. That scalar is exactly the sum of u_i times v_i, which is the dot product. This identity u · v = u^T v is the bridge between geometric dot products and matrix algebra, and it motivates the definition of the transpose in abstract inner product spaces.
Question 5 Short Answer
A student claims that (ABC)^T = A^T B^T C^T, reasoning that you just transpose each factor. What is the correct formula, and why does the student's version fail?
Think about your answer, then reveal below.
Model answer: (ABC)^T = C^T B^T A^T — the order of the factors reverses. The student's version fails because transposing a product is not the same as multiplying the transposed factors in the same order. Applying the rule twice: (ABC)^T = ((AB)C)^T = C^T (AB)^T = C^T B^T A^T. The reversal is necessary because matrix multiplication is not commutative, and the dimensions only line up correctly when the order is reversed.
The reversal rule (AB)^T = B^T A^T is analogous to (AB)^{-1} = B^{-1}A^{-1}: both arise because reversing the order is what makes dimensions (or the product with the identity) work out. For n factors, (A_1 A_2 ... A_n)^T = A_n^T ... A_2^T A_1^T. Students who forget this reversal will write expressions that are dimensionally impossible for non-square matrices, which is the easiest way to catch the error.