Question

Suppose the matrix product AB is defined.

(a) If B is a 85×11 matrix and A is a row matrix, give the dimensions of A and AB. (12 marks)
(b) Suppose Bᵀ−3B is defined and AB is a97×47 matrix. What is the size of matrix B ? (6 marks)
(c) If Bᵀ is a 20×81 matrix and (AB)ᵀ is a 20×19 matrix, what size is matrix A ? (6 marks)

Answer 1

Matrix A has dimensions 1×85 for AB to be defined with B being 85×11. If AB is 97×47, then B must be 97×11. If B⁴ is 20×81 and (AB)⁴ is 20×19, then A is 19×81.

Step-by-step explanation:

If the matrix product AB is defined and B is an 85×11 matrix, and A is a row matrix, then the dimensions of A must be 1×85, so the product AB will be a 1×11 matrix. The number of columns in A must match the number of rows in B for the matrix multiplication to be possible.

For part (b), if B⁴−3B is defined, this implies that B and B⁴ must have the same dimensions. Given that AB is a 97×47 matrix, A must have 97 rows. However, since B has 11 columns (from the information in part a), it means that the size of matrix B is 97×11.

For part (c), if B⁴ is a 20×81 matrix, and (AB)⁴ is a 20×19 matrix, then matrix AB must be a 19×20 matrix. Since A must be multiplied by B (which has 81 rows), A must have 19 rows and 81 columns.