Given $$A = \begin{pmatrix} 2 & 0 & 4 \\ 1 & 3 & 1 \end{pmatrix}$$​ and $$B = \begin{pmatrix} 4 & 5 & 0 \end{pmatrix}$$​, which of the following is true?

A. $$AB^T \ is \begin{pmatrix} 8 \\ 19 \end{pmatrix}$$​​
B. $$AB^T$$ is defined​
C. $$AB$$​ is not defined
D. $$BA$$​ is defined
E. $$BA^T \ is \begin{pmatrix} 8 & 19 \end{pmatrix}$$​​

Choose the correct answer from the options given below:

Correct option is B

Correct Option: 2. A, B, C & E

Explanation:

The question tests matrix multiplication rules and the properties of transposes.

• Statement A (True): $$AB^T$$​ results in a column vector $$\begin{pmatrix} 8 \\ 19 \end{pmatrix}$$​.

• Statement B (True): Since the inner dimensions match ($$A\ is\ 2 \times 3, B^T \ is\ 3 \times 1$$​), the product $$AB^T$$​is defined.

• Statement C (True): $$AB$$ requires multiplying $$(2 \times 3)$$ by $$(1 \times 3).$$​ The inner dimensions ($$3\ and \ 1$$​) do not match, so it is undefined.

• Statement E (True): $$BA^T$$​ involves $$(1 \times 3) \times (3 \times 2)$$​, which results in the row vector $$\begin{pmatrix} 8 & 19 \end{pmatrix}$$​.

Information Booster:

• Matrix Dimensions:

  • ​$$A: 2 \times 3$$​​
    

  • ​$$B: 1 \times 3$$​​
    

• Calculation for $$AB^T$$​:

  ​$$\begin{pmatrix} 2 & 0 & 4 \\ 1 & 3 & 1 \end{pmatrix} \begin{pmatrix} 4 \\ 5 \\ 0 \end{pmatrix} = \begin{pmatrix} (2)(4)+0+0 \\ (1)(4)+(3)(5)+0 \end{pmatrix} = \begin{pmatrix} 8 \\ 19 \end{pmatrix}$$​​
  

• Calculation for $$BA^T$$​:

  ​$$\begin{pmatrix} 4 & 5 & 0 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 0 & 3 \\ 4 & 1 \end{pmatrix} = \begin{pmatrix} 8 & 19 \end{pmatrix}$$​​
  

Additional Knowledge:

• Transpose Property: $$(AB)^T = B^T A^T$$​. This explains why the result of $$BA^T$$​ is the transpose of $$AB^T$$​(a row vector vs. a column vector).

• Compatibility: For matrix multiplication XY to be defined, the number of columns in X must equal the number of rows in Y.