Discussion
Let \(A = (a_{ij})_{3 \times 3}, B = (b_{ij})_{3 \times 2}\) and \(C = (c_{ij})_{3 \times 1}\).Which one of the following products, is not defined ?
Answer: D
The product that is not defined is \((AB)C\).
A product XY is defined when the number of columns of X equals the number of rows of Y. AB is \(3 \times 3\) times \(3 \times 2\), giving a \(3 \times 2\) matrix; multiplying it by the \(3 \times 1\) matrix C needs 2 = 3, which fails.
The others are defined: \(C^{T}AB\) is \((1 \times 3)(3 \times 3)(3 \times 2) = 1 \times 2\); \(A^{T}AB\) is \(3 \times 2\); \((AB)^{T}C\) is \((2 \times 3)(3 \times 1) = 2 \times 1\); and \(B^{T}C\) is \((2 \times 3)(3 \times 1) = 2 \times 1\).
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