Question

multiplying matrices (rehearse)

1. find the product: $\begin{bmatrix}-7&6\\\\1&6end{bmatrix}\begin{bmatrix}-4&1\\\\-4&3end{bmatrix}$

a) $\begin{bmatrix}4&11\\\\19&-28end{bmatrix}$
b) $\begin{bmatrix}4&11\\\\-28&19end{bmatrix}$
c) $\begin{bmatrix}28&-24\\\\-7&18end{bmatrix}$
d) $\begin{bmatrix}-4&-24\\\\1&18end{bmatrix}$

1. find the product of ab: $a = \begin{bmatrix}1&3&-3\\\\3&0&5end{bmatrix}$ $b = \begin{bmatrix}3&0\\\\-3&1\\\\0&5end{bmatrix}$

a) $\begin{bmatrix}-6&-12\\\\9&25end{bmatrix}$
b) $\begin{bmatrix}3&-9&0\\\\0&0&25end{bmatrix}$
c) ab is undefined
d) $\begin{bmatrix}-12&-6\\\\25&9end{bmatrix}$

1. find the product of cd: $c = \begin{bmatrix}3&-2&1\\\\-1&0&-5end{bmatrix}$ $d = \begin{bmatrix}4&-2\\\\2&3end{bmatrix}$

a) $\begin{bmatrix}14&-8&14\\\\3&-4&-13end{bmatrix}$
b) $\begin{bmatrix}8&-12\\\\-4&2end{bmatrix}$
c) $\begin{bmatrix}14&-8&14\\\\3&-4&-13end{bmatrix}$
d) cd is undefined

Explanation:

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Problem 1

Step1: Calculate top-left element

$(-7)(-4) + (6)(-4) = 28 - 24 = 4$

Step2: Calculate top-right element

$(-7)(1) + (6)(3) = -7 + 18 = 11$

Step3: Calculate bottom-left element

$(1)(-4) + (6)(-4) = -4 - 24 = -28$

Step4: Calculate bottom-right element

$(1)(1) + (6)(3) = 1 + 18 = 19$
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Problem 2

Step1: Check matrix dimensions

$A$ is $2 \times 3$, $B$ is $3 \times 2$. Columns of $A$ = Rows of $B$, so product exists.

Step2: Calculate top-left element

$(1)(3) + (3)(-3) + (-3)(0) = 3 - 9 + 0 = -6$

Step3: Calculate top-right element

$(1)(0) + (3)(1) + (-3)(5) = 0 + 3 - 15 = -12$

Step4: Calculate bottom-left element

$(3)(3) + (0)(-3) + (5)(0) = 9 + 0 + 0 = 9$

Step5: Calculate bottom-right element

$(3)(0) + (0)(1) + (5)(5) = 0 + 0 + 25 = 25$
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Problem 3

Step1: Check matrix dimensions

$C$ is $2 \times 3$, $D$ is $2 \times 2$. Columns of $C$ (3) ≠ Rows of $D$ (2), so product is undefined.
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Answer:

1. b)

$$\begin{bmatrix} 4 & 11 \\ -28 & 19 \end{bmatrix}$$

1. a)

$$\begin{bmatrix} -6 & -12 \\ 9 & 25 \end{bmatrix}$$

1. d) CD is undefined