Vector Multiplication (Matrices being Transformations of Vectors): CCSS.Math.Content.HSN-VM.C.11

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Common Core: High School - Number and Quantity › Vector Multiplication (Matrices being Transformations of Vectors): CCSS.Math.Content.HSN-VM.C.11

Questions 1 - 10

Calculate

$\begin{bmatrix} -10&19 \ -4& 16 \end{bmatrix} \times \begin{bmatrix} 11\ 0 \end{bmatrix}$

$\begin{bmatrix} -110\ -44 \end{bmatrix}$

$\begin{bmatrix} 110\ -44 \end{bmatrix}$

$\begin{bmatrix} -110\ 44 \end{bmatrix}$

$\begin{bmatrix} 110\ 44 \end{bmatrix}$

$\begin{bmatrix} -44\ -110 \end{bmatrix}$

Explanation

In order to do matrix multiplication, we need to check if the dimensions check out. The matrix on the left is $2\times2$ , and the matrix on the right is $2\times1$ , so the dimensions check out. The resulting matrix will be $2\times1$ . To do matrix multiplication, we take the rows of the matrix on the left, and multiply by the columns of the right matrix. Then we sum the results together. This is what it looks like in general.

$\begin{bmatrix} a& b\ c& d \end{bmatrix} \times \begin{bmatrix} e\ f \end{bmatrix} = \begin{bmatrix} a\cdot e+b\cdot f\ c\cdot e+d\cdot f \end{bmatrix}$

Now we apply the above to get the following solution.

$\begin{bmatrix} -10&19 \ -4& 16 \end{bmatrix} \times \begin{bmatrix} 11\ 0 \end{bmatrix}=\begin{bmatrix} -10\cdot11+19\cdot0\ -4\cdot 11+16\cdot0 \end{bmatrix} =\begin{bmatrix} -110+0\ -44+0 \end{bmatrix} =\begin{bmatrix} -110\ -44 \end{bmatrix}$

Calculate

$\begin{bmatrix} 17&14 \ 4& -19 \end{bmatrix} \times \begin{bmatrix} 10\ -16 \end{bmatrix}$

$\begin{bmatrix} -54\ 344 \end{bmatrix}$

$\begin{bmatrix} 54\ 344 \end{bmatrix}$

$\begin{bmatrix} -344\ 54 \end{bmatrix}$

$\begin{bmatrix} -54\ -344 \end{bmatrix}$

$\begin{bmatrix} 344\ 54 \end{bmatrix}$

Explanation

$\begin{bmatrix} a& b\ c& d \end{bmatrix} \times \begin{bmatrix} e\ f \end{bmatrix} = \begin{bmatrix} a\cdot e+b\cdot f\ c\cdot e+d\cdot f \end{bmatrix}$

Now we apply the above to get the following solution.

$\begin{bmatrix} 17&14 \ 4& -19 \end{bmatrix} \times \begin{bmatrix} 10\ -16 \end{bmatrix}=\begin{bmatrix} 17\cdot10+14\cdot-16\ 4\cdot 10+(-19)\cdot(-16) \end{bmatrix} =\begin{bmatrix} 170-224\ 40+304 \end{bmatrix} =\begin{bmatrix} -54\ 344 \end{bmatrix}$

Calculate

$\begin{bmatrix} 16&-20 \ -15& -1 \end{bmatrix} \times \begin{bmatrix} 15\ -2 \end{bmatrix}$

$\begin{bmatrix} 280\ -223 \end{bmatrix}$

$\begin{bmatrix} -280\ -223 \end{bmatrix}$

$\begin{bmatrix} -280\ 223 \end{bmatrix}$

$\begin{bmatrix} 280\ 223 \end{bmatrix}$

$\begin{bmatrix} -223\280 \end{bmatrix}$

Explanation

$\begin{bmatrix} a& b\ c& d \end{bmatrix} \times \begin{bmatrix} e\ f \end{bmatrix} = \begin{bmatrix} a\cdot e+b\cdot f\ c\cdot e+d\cdot f \end{bmatrix}$

Now we apply the above to get the following solution.

$\begin{bmatrix} 16&-20 \ -15& -1 \end{bmatrix} \times \begin{bmatrix} 15\ -2 \end{bmatrix}=\begin{bmatrix} 16\cdot15+(-20)\cdot(-2)\ (-15)\cdot 15+(-1)\cdot(-2) \end{bmatrix} =\begin{bmatrix} 240+40\ -225+2 \end{bmatrix} =\begin{bmatrix} 280\ -223 \end{bmatrix}$

Calculate

$\begin{bmatrix} 7&-1 \ -16& -1 \end{bmatrix} \times \begin{bmatrix} 17\ 3 \end{bmatrix}$

$\begin{bmatrix} 116\ -275 \end{bmatrix}$

$\begin{bmatrix} -116\ -275 \end{bmatrix}$

$\begin{bmatrix} 116\ 275 \end{bmatrix}$

$\begin{bmatrix} -275\ 116 \end{bmatrix}$

$\begin{bmatrix} -275\ -116 \end{bmatrix}$

Explanation

$\begin{bmatrix} a& b\ c& d \end{bmatrix} \times \begin{bmatrix} e\ f \end{bmatrix} = \begin{bmatrix} a\cdot e+b\cdot f\ c\cdot e+d\cdot f \end{bmatrix}$

Now we apply the above to get the following solution.

$\begin{bmatrix} 7&-1 \ -16& -1 \end{bmatrix} \times \begin{bmatrix} 17\ 3 \end{bmatrix}=\begin{bmatrix} 7\cdot17+(-1)\cdot3\ (-16)\cdot 17+(-1)\cdot3 \end{bmatrix} =\begin{bmatrix} 119-3\ -272-3 \end{bmatrix} =\begin{bmatrix} 116\ -275 \end{bmatrix}$

Calculate

$\begin{bmatrix} -5&8 \ -8& -5 \end{bmatrix} \times \begin{bmatrix} -15\ 10 \end{bmatrix}$

$\begin{bmatrix} 155\ 70 \end{bmatrix}$

$\begin{bmatrix} -155\ 70 \end{bmatrix}$

$\begin{bmatrix} 155\ -70 \end{bmatrix}$

$\begin{bmatrix} 70 \155\ \end{bmatrix}$

$\begin{bmatrix} -70 \155\ \end{bmatrix}$

Explanation

$\begin{bmatrix} a& b\ c& d \end{bmatrix} \times \begin{bmatrix} e\ f \end{bmatrix} = \begin{bmatrix} a\cdot e+b\cdot f\ c\cdot e+d\cdot f \end{bmatrix}$

Now we apply the above to get the following solution.

$\begin{bmatrix} -5&8 \ -8& -5 \end{bmatrix} \times \begin{bmatrix} -15\ 10 \end{bmatrix}=\begin{bmatrix} (-5)\cdot(-15)+8\cdot10\ (-8)\cdot (-15)+(-5)\cdot10 \end{bmatrix} =\begin{bmatrix} 75+80\ 120-50 \end{bmatrix} =\begin{bmatrix} 155\ 70 \end{bmatrix}$

Calculate

$\begin{bmatrix} -2&-19 \ 9& 5 \end{bmatrix} \times \begin{bmatrix} -12\ 10 \end{bmatrix}$

$\begin{bmatrix} -166\ -58 \end{bmatrix}$

$\begin{bmatrix} -166\ 58 \end{bmatrix}$

$\begin{bmatrix} 166\ -58 \end{bmatrix}$

$\begin{bmatrix} 166\ 58 \end{bmatrix}$

$\begin{bmatrix} -58\ -166 \end{bmatrix}$

Explanation

$\begin{bmatrix} a& b\ c& d \end{bmatrix} \times \begin{bmatrix} e\ f \end{bmatrix} = \begin{bmatrix} a\cdot e+b\cdot f\ c\cdot e+d\cdot f \end{bmatrix}$

Now we apply the above to get the following solution.

$\ \begin{bmatrix} -2&-19 \ 9& 5 \end{bmatrix} \times \begin{bmatrix} -12\ 10 \end{bmatrix}=\begin{bmatrix}(-2)\cdot(-12)+(-19)\cdot10\ 9\cdot (-12)+5\cdot10 \end{bmatrix} =\begin{bmatrix} 24-190\ -108+50 \end{bmatrix} =\begin{bmatrix} -166\ -58 \end{bmatrix}$

Calculate

$\begin{bmatrix} 8&-11 \ 9& 20 \end{bmatrix} \times \begin{bmatrix} -8\ -5 \end{bmatrix}$

$\begin{bmatrix} -9\ -172 \end{bmatrix}$

$\begin{bmatrix} 9\ -172 \end{bmatrix}$

$\begin{bmatrix} -9\ 172 \end{bmatrix}$

$\begin{bmatrix} 9\ 172 \end{bmatrix}$

$\begin{bmatrix} -172\ -9 \end{bmatrix}$

Explanation

$\begin{bmatrix} a& b\ c& d \end{bmatrix} \times \begin{bmatrix} e\ f \end{bmatrix} = \begin{bmatrix} a\cdot e+b\cdot f\ c\cdot e+d\cdot f \end{bmatrix}$

Now we apply the above to get the following solution.

$\begin{bmatrix} 8&-11 \ 9& 20 \end{bmatrix} \times \begin{bmatrix} -8\ -5 \end{bmatrix}=\begin{bmatrix} 8\cdot(-8)+(-11)\cdot(-5)\ 9\cdot (-8)+20\cdot(-5) \end{bmatrix} =\begin{bmatrix} -64+55\ -72-100 \end{bmatrix} =\begin{bmatrix} -9\ -172 \end{bmatrix}$

Calculate

$\begin{bmatrix} 18&-13 \ 6& -2 \end{bmatrix} \times \begin{bmatrix} 10\ -13 \end{bmatrix}$

$\begin{bmatrix} 349\ 86 \end{bmatrix}$

$\begin{bmatrix} -349\ 86 \end{bmatrix}$

$\begin{bmatrix} 349\ -86 \end{bmatrix}$

$\begin{bmatrix} -349\ -86 \end{bmatrix}$

$\begin{bmatrix} 86\349 \end{bmatrix}$

Explanation

$\begin{bmatrix} a& b\ c& d \end{bmatrix} \times \begin{bmatrix} e\ f \end{bmatrix} = \begin{bmatrix} a\cdot e+b\cdot f\ c\cdot e+d\cdot f \end{bmatrix}$

Now we apply the above to get the following solution.

$\begin{bmatrix} 18&-13 \ 6& -2 \end{bmatrix} \times \begin{bmatrix} 10\ -13 \end{bmatrix}=\begin{bmatrix} 18\cdot10+(-13)\cdot(-13)\ 6\cdot 10+(-2)\cdot(-13) \end{bmatrix} =\begin{bmatrix} 180+169\ 60+26 \end{bmatrix} =\begin{bmatrix} 349\ 86 \end{bmatrix}$

Calculate

$\begin{bmatrix} -4&6 \ -9& -8 \end{bmatrix} \times \begin{bmatrix} -6\ 4 \end{bmatrix}$

$\begin{bmatrix} 48\ 22 \end{bmatrix}$

$\begin{bmatrix} 48\ -22 \end{bmatrix}$

$\begin{bmatrix} -48\ 22 \end{bmatrix}$

$\begin{bmatrix} -48\ -22 \end{bmatrix}$

$\begin{bmatrix} 22\ -48 \end{bmatrix}$

Explanation

$\begin{bmatrix} a& b\ c& d \end{bmatrix} \times \begin{bmatrix} e\ f \end{bmatrix} = \begin{bmatrix} a\cdot e+b\cdot f\ c\cdot e+d\cdot f \end{bmatrix}$

Now we apply the above to get the following solution.

$\begin{bmatrix} -4&6 \ -9& -8 \end{bmatrix} \times \begin{bmatrix} -6\ 4 \end{bmatrix}=\begin{bmatrix} (-4)\cdot(-6)+6\cdot4\ (-9)\cdot (-6)+(-8)\cdot4 \end{bmatrix} =\begin{bmatrix} 24+24\ 54-32 \end{bmatrix} =\begin{bmatrix} 48\ 22 \end{bmatrix}$

Calculate

$\begin{bmatrix} -19&-17 \ 0& -5 \end{bmatrix} \times \begin{bmatrix} 12\ 19 \end{bmatrix}$

$\begin{bmatrix} -551\ -95 \end{bmatrix}$

$\begin{bmatrix} 551\ -95 \end{bmatrix}$

$\begin{bmatrix} -551\ 95 \end{bmatrix}$

$\begin{bmatrix} 551\ 95 \end{bmatrix}$

$\begin{bmatrix} -95 \-551\end{bmatrix}$

Explanation

$\begin{bmatrix} a& b\ c& d \end{bmatrix} \times \begin{bmatrix} e\ f \end{bmatrix} = \begin{bmatrix} a\cdot e+b\cdot f\ c\cdot e+d\cdot f \end{bmatrix}$

Now we apply the above to get the following solution.

$\begin{bmatrix} -19&-17 \ 0& -5 \end{bmatrix} \times \begin{bmatrix} 12\ 19 \end{bmatrix}=\begin{bmatrix} -19\cdot12+(-17)\cdot19\ 0\cdot 12+(-5)\cdot19 \end{bmatrix} =\begin{bmatrix} -228-323\ 0-95 \end{bmatrix} =\begin{bmatrix} -551\ -95 \end{bmatrix}$

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