Matrix A is 3×4 and matrix B is 4×2. What are the dimensions of the product AB?
A4×4
B3×2
C2×3
DUndefined — incompatible dimensions
For AB to be defined, the number of columns in A must equal the number of rows in B — here both are 4, so the product is defined. The result has the same number of rows as A (3) and the same number of columns as B (2), giving a 3×2 matrix. The rule: if A is m×n and B is n×p, then AB is m×p.
Question 2 True / False
For any two square matrices A and B of the same size, AB generally equals BA.
TTrue
FFalse
Answer: False
Matrix multiplication is not commutative. Even when both A and B are square and the same size (so both AB and BA are defined and have the same dimensions), AB ≠ BA in general. You can verify this with any two 2×2 matrices. The identity matrix is a special case where AI = IA = A, but this does not generalize.
Question 3 Short Answer
Why does the identity matrix have 1s on the diagonal and 0s elsewhere, rather than 1s everywhere?
Think about your answer, then reveal below.
Model answer: The identity matrix is defined by the property AI = IA = A. The diagonal 1s ensure that each row-column dot product returns the original matrix entry, while the off-diagonal 0s prevent mixing entries from different rows or columns. A matrix of all 1s would add up entries in a way that changes A.
The (i,j) entry of AI is the dot product of row i of A with column j of I. Column j of I has a 1 only in position j and 0s elsewhere, so the dot product returns the j-th entry of row i of A — exactly what's needed to leave A unchanged. All-1s would sum an entire row into each entry, which destroys the original matrix.