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Linear Algebra : Matrix-Vector Product

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Example Question #1 : Matrix Vector Product

Multiply: $\begin{bmatrix} 2 & 0& -5\\ -1& 3& 1\end{bmatrix} \cdot \begin{bmatrix} 2\\ 2\\ 1\end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 9\\5 \end{bmatrix}$

$\begin{bmatrix} -1\\8 \end{bmatrix}$

$\begin{bmatrix} 9\\8 \end{bmatrix}$

$\begin{bmatrix} -1\\5 \end{bmatrix}$

$\begin{bmatrix} -1\\ -3\end{bmatrix}$

Correct answer:

$\begin{bmatrix} -1\\5 \end{bmatrix}$

Explanation:

To multiply, add:

$2 \begin{bmatrix} 2\\ -1\end{bmatrix} + 2 \begin{bmatrix} 0\\3 \end{bmatrix} + 1 \begin{bmatrix} -5\\ 1\end{bmatrix}$

$= \begin{bmatrix} 4\\-2 \end{bmatrix} + \begin{bmatrix} 0\\ 6\end{bmatrix} + \begin{bmatrix} -5\\ 1\end{bmatrix} = \begin{bmatrix} -1\\ 5\end{bmatrix}$

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Example Question #1 : Matrix Vector Product

Compute AB.

$A=\begin{bmatrix} 1&3 \\2 &5 \end{bmatrix}$

$B = \begin{bmatrix} 4& 1& 3\\ 6& -2&0 \end{bmatrix}$

Possible Answers:

$AB = \begin{bmatrix} 4& 3&3 \\ 12&-10 &0 \end{bmatrix}$

$AB = \begin{bmatrix} 5&4 &3 \\ 8&3 &0 \end{bmatrix}$

$AB = \begin{bmatrix} -14&7 & 3\\ -22& 12&6 \end{bmatrix}$

$AB = \begin{bmatrix} 22& -5&3 \\ 38&-8 &6 \end{bmatrix}$

None of the other answers.

Correct answer:

$AB = \begin{bmatrix} 22& -5&3 \\ 38&-8 &6 \end{bmatrix}$

Explanation:

Because the number of columns in matrix A and the number of rows in matrix B are equal, we know that product AB does in fact exist. Matrix AB should have the same number of rows as A and the same number of columns as B. In this case, AB is a 2x3 matrix:

$AB=\begin{bmatrix} (1\cdot 4+3\cdot 6) & (1\cdot 1+3\cdot -2) &(1\cdot 3+3\cdot 0) \\ (2\cdot 4+5\cdot 6) & (2\cdot 1+5\cdot -2) &(2\cdot 3+5\cdot 0) \end{bmatrix}$

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Example Question #3 : Matrix Vector Product

Compute AB

$A=\begin{bmatrix} 5\end{bmatrix}$

$B = \begin{bmatrix} -5&8 &12 &7 \end{bmatrix}$

Possible Answers:

None of the other answers

$AB = \begin{bmatrix} -25\\40 \\60 \\35 \end{bmatrix}$

$AB = \begin{bmatrix} 0\\ 13\\17 \\12 \end{bmatrix}$

$AB= \begin{bmatrix} -25&40 &60 &35 \end{bmatrix}$

$AB = \begin{bmatrix} 0&13 &17 &12 \end{bmatrix}$

Correct answer:

$AB= \begin{bmatrix} -25&40 &60 &35 \end{bmatrix}$

Explanation:

$AB = \begin{bmatrix} 5\cdot -5& 5\cdot 8&5\cdot 12 &5\cdot 7 \end{bmatrix}$

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Example Question #4 : Matrix Vector Product

Calculate $A\bf{x}$ , given

$A=\begin{bmatrix} 3& 4\\ 0&-2 \\ 1&3 \end{bmatrix}$ , $\bf{x}=\begin{bmatrix} 6\\ 2 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 24\\ -4\\ 6 \end{bmatrix}$

$\begin{bmatrix} 26\\ -4\\ 12 \end{bmatrix}$

$\begin{bmatrix} 8\\ 0\\ 6 \end{bmatrix}$

$\begin{bmatrix} 18\\ 0\\ 6 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 26\\ -4\\ 12 \end{bmatrix}$

Explanation:

By definition,

$A\bf{x}=\begin{bmatrix} 3&4 \\ 0&-2 \\ 1& 3 \end{bmatrix} \begin{bmatrix} 6\\ 2 \end{bmatrix}= \begin{bmatrix} 3(6)+4(2)\\ 0(6)-2(2)\\ 1(6)+3(2) \end{bmatrix}= \begin{bmatrix} 26\\ -4\\ 12 \end{bmatrix}$ .

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Example Question #1 : Matrix Vector Product

Calculate , given

$A = \begin{bmatrix} 1&4 \end{bmatrix}$

$B=\begin{bmatrix} 3\\ 4 \end{bmatrix}$

Possible Answers:

$AB = \begin{bmatrix} 1& 4\\ 3& 4 \end{bmatrix}$

AB = 19

AB= 16

$AB= \begin{bmatrix} 3& 4\\ 1& 4 \end{bmatrix}$

Correct answer:

AB = 19

Explanation:

By definition,

$AB = \begin{bmatrix} 1& 4 \end{bmatrix} \begin{bmatrix} 3\\ 4 \end{bmatrix} = [3(1)+4(4)]= [19]=19$ . A matrix with only one entry is simply a scalar.

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Example Question #1 : Matrix Vector Product

Calculate , given

c = 3

$A= \begin{bmatrix} 3&1 \\ 5&-2 \end{bmatrix}$ .

Possible Answers:

cA = 21

$cA = \begin{bmatrix} 9&1 \\ 5&-6 \end{bmatrix}$

Can not be determined, is not a vector.

$cA = \begin{bmatrix} 9&3 \\ 15& -6 \end{bmatrix}$

Correct answer:

$cA = \begin{bmatrix} 9&3 \\ 15& -6 \end{bmatrix}$

Explanation:

By definition,

$cA = 3\begin{bmatrix} 3&1 \\ 5& -2 \end{bmatrix}= \begin{bmatrix} 3(3)&3(1) \\ 3(5)&3(-2) \end{bmatrix}= \begin{bmatrix} 9&3 \\ 15&-6 \end{bmatrix}$ .

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Example Question #1 : Matrix Vector Product

$\begin{align*}&\text{Find the product }A\times B \\&\text{Where }A=\begin{bmatrix}3&10\\3&-10\end{bmatrix}\text{ and }B=\begin{bmatrix}2&-8\\11&-19\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}116&-214\\-104&166\end{bmatrix}$

$\begin{bmatrix}182&-148\\-38&232\end{bmatrix}$

$\text{The multiplication cannot be performed.}$

$\begin{bmatrix}85&-245\\-135&135\end{bmatrix}$

Correct answer:

$\begin{bmatrix}116&-214\\-104&166\end{bmatrix}$

Explanation:

$\begin{align*}&\text{In order to multiply two matrices, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }m\times n\text{ and }n\times p\text{ to create an}\\&m\times p\text{ matrix. Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }2\times2\\&\text{and B has dimensions: }2\times2\\&\text{The resultant matrix has dimensions: }2\times2\\&\begin{bmatrix}3&10\\3&-10\end{bmatrix}\times\begin{bmatrix}2&-8\\11&-19\end{bmatrix}\\&\begin{bmatrix}(3)(2)+(10)(11)&(3)(-8)+(10)(-19)\\(3)(2)+(-10)(11)&(3)(-8)+(-10)(-19)\end{bmatrix}\\&\begin{bmatrix}116&-214\\-104&166\end{bmatrix}\end{align*}$

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Example Question #1 : Matrix Vector Product

$\begin{align*}&\text{Find the value of }C\text{ if }C=A\times B \\&\text{Where }A=\begin{bmatrix}-1&-12&-13\\-12&14&-11\\19&-7&1\end{bmatrix}\text{ and }B=\begin{bmatrix}-17\\-19\\-12\end{bmatrix}\end{align*}$

Possible Answers:

$\text{The multiplication cannot be performed.}$

$\begin{bmatrix}401\\70\\-202\end{bmatrix}$

$\begin{bmatrix}467\\136\\-136\end{bmatrix}$

$\begin{bmatrix}400\\69\\-203\end{bmatrix}$

Correct answer:

$\begin{bmatrix}401\\70\\-202\end{bmatrix}$

Explanation:

$\begin{align*}&\text{In order to multiply two matrices, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }m\times n\text{ and }n\times p\text{ to create an}\\&m\times p\text{ matrix. Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }3\times3\\&\text{and B has dimensions: }3\times1\\&\text{The resultant matrix C has dimensions: }3\times1\\&C=\begin{bmatrix}-1&-12&-13\\-12&14&-11\\19&-7&1\end{bmatrix}\times\begin{bmatrix}-17\\-19\\-12\end{bmatrix}\\&C=\begin{bmatrix}(-1)(-17)+(-12)(-19)+(-13)(-12)\\(-12)(-17)+(14)(-19)+(-11)(-12)\\(19)(-17)+(-7)(-19)+(1)(-12)\end{bmatrix}\\&C=\begin{bmatrix}401\\70\\-202\end{bmatrix}\end{align*}$

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Example Question #9 : Matrix Vector Product

$\begin{align*}&\text{Find the matrix product }\\&\begin{bmatrix}-19&4\\-6&-12\\-18&4\end{bmatrix}\times\begin{bmatrix}-10\\-19\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}181\\355\\171\end{bmatrix}$

$\begin{bmatrix}114\\288\\104\end{bmatrix}$

$\text{The multiplication cannot be performed.}$

$\begin{bmatrix}130\\304\\120\end{bmatrix}$

Correct answer:

$\begin{bmatrix}114\\288\\104\end{bmatrix}$

Explanation:

$\begin{align*}&\text{In order to multiply two matrices, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }m\times n\text{ and }n\times p\text{ to create an}\\&m\times p\text{ matrix. Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since our matrices have dimensions: }3\times2\\&\text{and }2\times1\\&\text{The resultant matrix has dimensions: }3\times1\\&\begin{bmatrix}-19&4\\-6&-12\\-18&4\end{bmatrix}\times\begin{bmatrix}-10\\-19\end{bmatrix}\\&\begin{bmatrix}(-19)(-10)+(4)(-19)\\(-6)(-10)+(-12)(-19)\\(-18)(-10)+(4)(-19)\end{bmatrix}\\&\begin{bmatrix}114\\288\\104\end{bmatrix}\end{align*}$

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Example Question #10 : Matrix Vector Product

$\begin{align*}&\text{Find the product }A\times B \\&\text{Where }A=\begin{bmatrix}17&7&-7&2\\10&0&12&-2\\-8&-4&20&-3\end{bmatrix}\text{ and }B=\begin{bmatrix}5\\20\\-19\\-2\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}354\\-174\\-494\end{bmatrix}$

$\begin{bmatrix}356\\-172\\-492\end{bmatrix}$

$\text{The multiplication cannot be performed.}$

$\begin{bmatrix}433\\-95\\-415\end{bmatrix}$

Correct answer:

$\begin{bmatrix}354\\-174\\-494\end{bmatrix}$

Explanation:

$\begin{align*}&\text{In order to multiply two matrices, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }m\times n\text{ and }n\times p\text{ to create an}\\&m\times p\text{ matrix. Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }3\times4\\&\text{and B has dimensions: }4\times1\\&\text{The resultant matrix has dimensions: }3\times1\\&\begin{bmatrix}17&7&-7&2\\10&0&12&-2\\-8&-4&20&-3\end{bmatrix}\times\begin{bmatrix}5\\20\\-19\\-2\end{bmatrix}\\&\begin{bmatrix}(17)(5)+(7)(20)+(-7)(-19)+(2)(-2)\\(10)(5)+(0)(20)+(12)(-19)+(-2)(-2)\\(-8)(5)+(-4)(20)+(20)(-19)+(-3)(-2)\end{bmatrix}\\&\begin{bmatrix}354\\-174\\-494\end{bmatrix}\end{align*}$

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Linear Algebra : Matrix-Vector Product

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #1 : Matrix Vector Product

Example Question #1 : Matrix Vector Product

Example Question #3 : Matrix Vector Product

Example Question #4 : Matrix Vector Product

Example Question #1 : Matrix Vector Product

Example Question #1 : Matrix Vector Product

Example Question #1 : Matrix Vector Product

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