Matrix Multiplication

**pepelepew23** · August 6th 2012, 02:56 PM

Hello,

What is the product of the following two matrices? In my book, the answer is A, while I believe it is B because if c₁= r₂, then a product exists. However, this is not upheld in this given question because c₁= 1, and r₂= 4.
Here is the problem:
The product of (-2 -1 5 9) (0 -5)
(3 -2)
(4 0)
(-7 1) is:

Answers~
A) (-37)
(21)
B) product is not defined
The other choices are irrelevant to my question.
Thank you for the help, it is greatly appreciated.

**Soroban** · August 6th 2012, 03:51 PM

Hello, pepelepew23!

What is the product of the following two matrices?

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

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

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

In my book, the answer is A, while I believe it is B
. . because if c₁= r₂, then a product exists.
However, this is not upheld in this given question because c₁= 1, and r₂= 4.
Not true . . . The first matrix has four columns: c₁ = 4.

We are multiplying a $1\!\times\!4$ matrix by a $4\!\times\!2$ matrix.
The product is a $1\!\times\!2$ matrix . . . That's one row and two columns: . $\begin{pmatrix}* & * \end{pmatrix}$
The book's answer is wrong; it is a $2\!\times\!1$ matrix . . . the wrong shape!

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

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

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

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

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