Re: Matrix Multiplication

Hello, pepelepew23!

Quote:

What is the product of the following two matrices?

. . $\displaystyle \begin{pmatrix}\text{-}2 & \text{-}1 & 5 & 9 \end{pmatrix}\begin{pmatrix}0 & \text{-}5 \\ 3 & \text{-}2 \\ 4 & 0 \\ \text{-}7 & 1 \end{pmatrix}$

. . $\displaystyle (A)\;\begin{pmatrix}\text{-}37 \\ 21 \end{pmatrix}\qquad (B)\;\text{product is not defined.}$

The other choices are irrelevant to my question. . Really?

In my book, the answer is A, while I believe it is B

. . because if c_{1 }= r_{2 }, then a product exists.

However, this is not upheld in this given question because c_{1 }= 1, and r_{2}= 4.

Not true . . . The first matrix has **four** columns: c_{1} = 4.

We are multiplying a $\displaystyle 1\!\times\!4$ matrix by a $\displaystyle 4\!\times\!2$ matrix.

The product is a $\displaystyle 1\!\times\!2$ matrix . . . That's **one row** and **two columns**: .$\displaystyle \begin{pmatrix}* & * \end{pmatrix}$

The book's answer is wrong; it is a $\displaystyle 2\!\times\!1$ matrix . . . the wrong shape!

$\displaystyle \begin{pmatrix}\text{-}2 & \text{-}1 & 5 & 9 \end{pmatrix}\begin{pmatrix}0 & \text{-}5 \\ 3 & \text{-}2 \\ 4 & 0 \\ \text{-}7 & 1 \end{pmatrix}$

. . $\displaystyle =\;\bigg((\text{-}2)(0) + (\text{-}1)(3) + (5)(4) + (9)(\text{-}7),\;(\text{-}2)(\text{-}5) + (\text{-}1)(\text{-}2) + (5)(0) + (9)(1)\bigg) $

. . $\displaystyle =\;\(0-3+20-63,\:10 + 2 + 0 + 9)$

. . $\displaystyle =\;(\text{-}46,\:21)$