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Linear Algebra : Matrices

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Example Questions

Example Question #2 : 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:

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

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

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

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

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 #1 : 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:

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

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

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

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

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 #3 : 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:

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

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

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

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

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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Example Question #111 : Matrices

$\begin{align*}&\text{Find the matrix product }\\&\begin{bmatrix}9&-11\\14&-3\end{bmatrix}\times\begin{bmatrix}7\\11\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}-33\\90\end{bmatrix}$

$\begin{bmatrix}-58\\65\end{bmatrix}$

$\begin{bmatrix}-124\\-1\end{bmatrix}$

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

Correct answer:

$\begin{bmatrix}-58\\65\end{bmatrix}$

Explanation:

$\begin{align*}&\text{In order to multiply two matrices, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }m\times n\text{ and }n\times p\text{ to create an}\\&m\times p\text{ matrix. Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since our matrices have dimensions: }2\times2\text{ and }2\times1\\&\text{The resultant matrix has dimensions: }2\times1\\&\begin{bmatrix}9&-11\\14&-3\end{bmatrix}\times\begin{bmatrix}7\\11\end{bmatrix}\\&\begin{bmatrix}(9)(7)+(-11)(11)\\(14)(7)+(-3)(11)\end{bmatrix}\\&\begin{bmatrix}-58\\65\end{bmatrix}\end{align*}$

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Example Question #112 : Matrices

$\begin{align*}&\text{Find the value of }C\text{ if }C=A\times B \\&\text{Where }A=\begin{bmatrix}-11&-1&0&2\\-16&8&0&-5\\-16&3&-18&-3\\0&8&-15&0\end{bmatrix}\text{ and }B=\begin{bmatrix}-15\\8\\-11\\-8\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}233\\436\\578\\321\end{bmatrix}$

$\begin{bmatrix}132\\335\\477\\220\end{bmatrix}$

$\begin{bmatrix}141\\344\\486\\229\end{bmatrix}$

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

Correct answer:

$\begin{bmatrix}141\\344\\486\\229\end{bmatrix}$

Explanation:

$\begin{align*}&\text{In order to multiply two matrices, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }m\times n\text{ and }n\times p\text{ to create an}\\&m\times p\text{ matrix. Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }4\times4\\&\text{and B has dimensions: }4\times1\\&\text{The resultant matrix C has dimensions: }4\times1\\&C=\begin{bmatrix}-11&-1&0&2\\-16&8&0&-5\\-16&3&-18&-3\\0&8&-15&0\end{bmatrix}\times\begin{bmatrix}-15\\8\\-11\\-8\end{bmatrix}\\&C=\begin{bmatrix}(-11)(-15)+(-1)(8)+(0)(-11)+(2)(-8)\\(-16)(-15)+(8)(8)+(0)(-11)+(-5)(-8)\\(-16)(-15)+(3)(8)+(-18)(-11)+(-3)(-8)\\(0)(-15)+(8)(8)+(-15)(-11)+(0)(-8)\end{bmatrix}\\&C=\begin{bmatrix}141\\344\\486\\229\end{bmatrix}\end{align*}$

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Example Question #113 : Matrices

$\begin{align*}&\text{Find the product }A\times B \\&\text{Where }A=\begin{bmatrix}-15&16&13&-5\\10&1&6&-6\end{bmatrix}\text{ and }B=\begin{bmatrix}19\\6\\20\\0\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}-14\\231\end{bmatrix}$

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

$\begin{bmatrix}100\\345\end{bmatrix}$

$\begin{bmatrix}71\\316\end{bmatrix}$

Correct answer:

$\begin{bmatrix}71\\316\end{bmatrix}$

Explanation:

$\begin{align*}&\text{In order to multiply two matrices, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }m\times n\text{ and }n\times p\text{ to create an}\\&m\times p\text{ matrix. Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }2\times4\\&\text{and B has dimensions: }4\times1\\&\text{The resultant matrix has dimensions: }2\times1\\&\begin{bmatrix}-15&16&13&-5\\10&1&6&-6\end{bmatrix}\times\begin{bmatrix}19\\6\\20\\0\end{bmatrix}\\&\begin{bmatrix}(-15)(19)+(16)(6)+(13)(20)+(-5)(0)\\(10)(19)+(1)(6)+(6)(20)+(-6)(0)\end{bmatrix}\\&\begin{bmatrix}71\\316\end{bmatrix}\end{align*}$

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Example Question #114 : Matrices

$\begin{align*}&\text{Find the matrix product }\\&\begin{bmatrix}2&11&13\\-1&-3&16\\-13&17&-7\\10&-12&-13\end{bmatrix}\times\begin{bmatrix}-11\\-3\\2\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}-29\\52\\78\\-100\end{bmatrix}$

$\begin{bmatrix}8\\89\\115\\-63\end{bmatrix}$

$\begin{bmatrix}71\\152\\178\\0\end{bmatrix}$

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

Correct answer:

$\begin{bmatrix}-29\\52\\78\\-100\end{bmatrix}$

Explanation:

$\begin{align*}&\text{In order to multiply two matrices, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }m\times n\text{ and }n\times p\text{ to create an}\\&m\times p\text{ matrix. Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since our matrices have dimensions: }4\times3\text{ and }3\times1\\&\text{The resultant matrix has dimensions: }4\times1\\&\begin{bmatrix}2&11&13\\-1&-3&16\\-13&17&-7\\10&-12&-13\end{bmatrix}\times\begin{bmatrix}-11\\-3\\2\end{bmatrix}\\&\begin{bmatrix}(2)(-11)+(11)(-3)+(13)(2)\\(-1)(-11)+(-3)(-3)+(16)(2)\\(-13)(-11)+(17)(-3)+(-7)(2)\\(10)(-11)+(-12)(-3)+(-13)(2)\end{bmatrix}\\&\begin{bmatrix}-29\\52\\78\\-100\end{bmatrix}\end{align*}$

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Example Question #115 : Matrices

$\begin{align*}&\text{Find the value of }C\text{ if }C=A\times B \\&\text{Where }A=\begin{bmatrix}10&0&-11\\-16&12&17\\-20&-20&-2\\17&5&6\end{bmatrix}\text{ and }B=\begin{bmatrix}6\\-18\\-17\end{bmatrix}\end{align*}$

Possible Answers:

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

$\begin{bmatrix}247\\-601\\274\\-90\end{bmatrix}$

$\begin{bmatrix}253\\-595\\280\\-84\end{bmatrix}$

$\begin{bmatrix}343\\-505\\370\\6\end{bmatrix}$

Correct answer:

$\begin{bmatrix}247\\-601\\274\\-90\end{bmatrix}$

Explanation:

$\begin{align*}&\text{In order to multiply two matrices, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }m\times n\text{ and }n\times p\text{ to create an}\\&m\times p\text{ matrix. Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }4\times3\\&\text{and B has dimensions: }3\times1\\&\text{The resultant matrix C has dimensions: }4\times1\\&C=\begin{bmatrix}10&0&-11\\-16&12&17\\-20&-20&-2\\17&5&6\end{bmatrix}\times\begin{bmatrix}6\\-18\\-17\end{bmatrix}\\&C=\begin{bmatrix}(10)(6)+(0)(-18)+(-11)(-17)\\(-16)(6)+(12)(-18)+(17)(-17)\\(-20)(6)+(-20)(-18)+(-2)(-17)\\(17)(6)+(5)(-18)+(6)(-17)\end{bmatrix}\\&C=\begin{bmatrix}247\\-601\\274\\-90\end{bmatrix}\end{align*}$

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Example Question #116 : Matrices

$\begin{align*}&\text{Find the value of }C\text{ if }C=A\times B \\&\text{Where }A=\begin{bmatrix}-7&15\\3&-8\end{bmatrix}\text{ and }B=\begin{bmatrix}16\\4\end{bmatrix}\end{align*}$

Possible Answers:

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

$\begin{bmatrix}-52\\16\end{bmatrix}$

$\begin{bmatrix}-19\\49\end{bmatrix}$

$\begin{bmatrix}-115\\-47\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-52\\16\end{bmatrix}$

Explanation:

$\begin{align*}&\text{In order to multiply two matrices, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }m\times n\text{ and }n\times p\text{ to create an}\\&m\times p\text{ matrix. Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since A has dimensions: }2\times2\\&\text{and B has dimensions: }2\times1\\&\text{The resultant matrix C has dimensions: }2\times1\\&C=\begin{bmatrix}-7&15\\3&-8\end{bmatrix}\times\begin{bmatrix}16\\4\end{bmatrix}\\&C=\begin{bmatrix}(-7)(16)+(15)(4)\\(3)(16)+(-8)(4)\end{bmatrix}\\&C=\begin{bmatrix}-52\\16\end{bmatrix}\end{align*}$

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Example Question #117 : Matrices

$\begin{align*}&\text{Find the matrix product }\\&\begin{bmatrix}-16&16\\3&19\\3&-1\\7&-13\end{bmatrix}\times\begin{bmatrix}-1\\-20\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}-402\\-481\\-81\\155\end{bmatrix}$

$\begin{bmatrix}-267\\-346\\54\\290\end{bmatrix}$

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

$\begin{bmatrix}-304\\-383\\17\\253\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-304\\-383\\17\\253\end{bmatrix}$

Explanation:

$\begin{align*}&\text{In order to multiply two matrices, }A\times b\text{, the respective dimensions of each}\\&\text{must be of the form }m\times n\text{ and }n\times p\text{ to create an}\\&m\times p\text{ matrix. Note that order does matter:}\\&(A\times b \neq b \times A)\\&\text{Since our matrices have dimensions: }4\times2\text{ and }2\times1\\&\text{The resultant matrix has dimensions: }4\times1\\&\begin{bmatrix}-16&16\\3&19\\3&-1\\7&-13\end{bmatrix}\times\begin{bmatrix}-1\\-20\end{bmatrix}\\&\begin{bmatrix}(-16)(-1)+(16)(-20)\\(3)(-1)+(19)(-20)\\(3)(-1)+(-1)(-20)\\(7)(-1)+(-13)(-20)\end{bmatrix}\\&\begin{bmatrix}-304\\-383\\17\\253\end{bmatrix}\end{align*}$

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Linear Algebra : Matrices

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #2 : Matrix Vector Product

Example Question #1 : Matrix Vector Product

Example Question #3 : Matrix Vector Product

Example Question #111 : Matrices

Example Question #112 : Matrices

Example Question #113 : Matrices

Example Question #114 : Matrices

Example Question #115 : Matrices

Example Question #116 : Matrices

Example Question #117 : Matrices

All Linear Algebra Resources

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