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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #52 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 1& 0\\5 &0 \end{vmatrix}$ and $b=\begin{vmatrix} 3\\0 \end{vmatrix}$ .

Possible Answers:

$\begin{vmatrix}15\\0\end{vmatrix}$

$\begin{vmatrix}3\\0\end{vmatrix}$

$\begin{vmatrix}3\\3\end{vmatrix}$

$\begin{vmatrix}15\\3 \end{vmatrix}$

$\begin{vmatrix}3\\15 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}3\\15 \end{vmatrix}$

Explanation:

In order to multiply two matrices, $A\times b$ , the respective dimensions of each must be of the form $m \times n$ and $n \times p$ to create an $m\times p$ (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

$(A\times b \neq b \times A)$

For a multiplication of the form

$\begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}$

The resulting matrix is

$\begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}$

The notation may be daunting but numerical examples may elucidate.

We're told that $A=\begin{vmatrix} 1& 0\\5 &0 \end{vmatrix}$ and $b=\begin{vmatrix} 3\\0 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}3+0\\15+0 \end{vmatrix}=\begin{vmatrix}3\\15 \end{vmatrix}$

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Example Question #58 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 1&2 \\11 &0 \end{vmatrix}$ and $b=\begin{vmatrix} 5\\2 \end{vmatrix}$ .

Possible Answers:

$\begin{vmatrix}99\\0 \end{vmatrix}$

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

$\begin{vmatrix}99\\55 \end{vmatrix}$

$\begin{vmatrix}9\\55 \end{vmatrix}$

$\begin{vmatrix}99\\5 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}9\\55 \end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that $A=\begin{vmatrix} 1&2 \\11 &0 \end{vmatrix}$ and $b=\begin{vmatrix} 5\\2 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}5+4\\55+0 \end{vmatrix}=\begin{vmatrix}9\\55 \end{vmatrix}$

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Example Question #53 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 7& 1\\3 &-3 \end{vmatrix}$ and $b=\begin{vmatrix} 4\\1 \end{vmatrix}$ .

Possible Answers:

$\begin{vmatrix}27\\ 15\end{vmatrix}$

$\begin{vmatrix}29\\ 9\end{vmatrix}$

$\begin{vmatrix}29\\ 15\end{vmatrix}$

$\begin{vmatrix}15\\ 29\end{vmatrix}$

$\begin{vmatrix}27\\ 9\end{vmatrix}$

Correct answer:

$\begin{vmatrix}29\\ 9\end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that $A=\begin{vmatrix} 7& 1\\3 &-3 \end{vmatrix}$ and $b=\begin{vmatrix} 4\\1 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}28+1\\12-3 \end{vmatrix}=\begin{vmatrix}29\\ 9\end{vmatrix}$

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Example Question #54 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 1&2 &4 \\0 &3 & 0\\5 &1 &-1 \end{vmatrix}$ and $b=\begin{vmatrix} 2\\0 \\3 \end{vmatrix}$ .

Possible Answers:

$\begin{vmatrix}14\\7 \\0 \end{vmatrix}$

$\begin{vmatrix}7\\14 \\0 \end{vmatrix}$

$\begin{vmatrix}14\\0 \\7 \end{vmatrix}$

$\begin{vmatrix}7\\0 \\14 \end{vmatrix}$

$\begin{vmatrix}0\\7 \\14 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}14\\0 \\7 \end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that $A=\begin{vmatrix} 1&2 &4 \\0 &3 & 0\\5 &1 &-1 \end{vmatrix}$ and $b=\begin{vmatrix} 2\\0 \\3 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}2+0+12\\0+0+0 \\10+0-3 \end{vmatrix}=\begin{vmatrix}14\\0 \\7 \end{vmatrix}$

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Example Question #61 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix}1 &3 &4 \\0 &2 & -1\\ 0& 0& 5\end{vmatrix}$ and $b=\begin{vmatrix} 0\\4 \\2 \end{vmatrix}$ .

Possible Answers:

$\begin{vmatrix}10\\6 \\20 \end{vmatrix}$

$\begin{vmatrix}20\\6 \\10 \end{vmatrix}$

$\begin{vmatrix}16\\6 \\20 \end{vmatrix}$

$\begin{vmatrix}10\\6 \\10 \end{vmatrix}$

$\begin{vmatrix}10\\26 \\20 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}20\\6 \\10 \end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that $A=\begin{vmatrix}1 &3 &4 \\0 &2 & -1\\ 0& 0& 5\end{vmatrix}$ and $b=\begin{vmatrix} 0\\4 \\2 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}0+12+8\\0+8-2 \\0+0+10 \end{vmatrix}=\begin{vmatrix}20\\6 \\10 \end{vmatrix}$

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Example Question #62 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 1&0 &0 \\0 &1 &0 \\0 &0 &1 \end{vmatrix}$ and $b=\begin{vmatrix} 4\\11 \\3 \end{vmatrix}$ .

Possible Answers:

$\begin{vmatrix}4\\11 \\3 \end{vmatrix}$

$\begin{vmatrix}44\\121 \\33 \end{vmatrix}$

$\begin{vmatrix}3\\4 \\11 \end{vmatrix}$

$\begin{vmatrix}1\\1 \\1 \end{vmatrix}$

$\begin{vmatrix}0\\0 \\0 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}4\\11 \\3 \end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that $A=\begin{vmatrix} 1&0 &0 \\0 &1 &0 \\0 &0 &1 \end{vmatrix}$ and $b=\begin{vmatrix} 4\\11 \\3 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}4+0+0\\ 0+11+0\\0+0+3 \end{vmatrix}=\begin{vmatrix}4\\11 \\3 \end{vmatrix}$

In this case, A is a special type of matrix known as an identity matrix.

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Example Question #63 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 11&2 \\2 &11 \end{vmatrix}$ and $b=\begin{vmatrix} 1\\3 \end{vmatrix}$ .

Possible Answers:

$\begin{vmatrix}17\\17 \end{vmatrix}$

$\begin{vmatrix}22\\66 \end{vmatrix}$

$\begin{vmatrix}35\\17 \end{vmatrix}$

$\begin{vmatrix}35\\35 \end{vmatrix}$

$\begin{vmatrix}17\\35 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}17\\35 \end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that $A=\begin{vmatrix} 11&2 \\2 &11 \end{vmatrix}$ and $b=\begin{vmatrix} 1\\3 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}11+6\\2+33 \end{vmatrix}=\begin{vmatrix}17\\35 \end{vmatrix}$

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Example Question #64 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 8& -3\\ -2&-4 \end{vmatrix}$ and $b=\begin{vmatrix} -1\\-2 \end{vmatrix}$ .

Possible Answers:

$\begin{vmatrix}24\\-16 \end{vmatrix}$

$\begin{vmatrix}2\\10 \end{vmatrix}$

$\begin{vmatrix}-14\\10 \end{vmatrix}$

$\begin{vmatrix}-2\\10 \end{vmatrix}$

$\begin{vmatrix}-14\\-10 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}-2\\10 \end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that $A=\begin{vmatrix} 8& -3\\ -2&-4 \end{vmatrix}$ and $b=\begin{vmatrix} -1\\-2 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}-8+6\\ 2+8\end{vmatrix}=\begin{vmatrix}-2\\10 \end{vmatrix}$

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Example Question #651 : Vectors And Vector Operations

Find the determinant of the matrix $A=\begin{vmatrix} 15&18 \\1 &17 \end{vmatrix}$

Possible Answers:

135

291

237

129

273

Correct answer:

237

Explanation:

The determinant of a 2x2 matrix can be found by cross multiplying terms as follows:

$det\begin{vmatrix} a&b \\ c&d \end{vmatrix}=ad-bc$

For the matrix $A=\begin{vmatrix} 15&18 \\1 &17 \end{vmatrix}$

The determinant is thus:

|A|=255-18=237

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Example Question #62 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix}20 & 27\\1 &15 \end{vmatrix}$

Possible Answers:

159

273

132

105

327

Correct answer:

273

Explanation:

The determinant of a 2x2 matrix can be found by cross multiplying terms as follows:

$det\begin{vmatrix} a&b \\ c&d \end{vmatrix}=ad-bc$

For the matrix $A=\begin{vmatrix}20 & 27\\1 &15 \end{vmatrix}$

The determinant is thus:

|A|=300-27=273

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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #52 : Matrices

Example Question #58 : Matrices

Example Question #53 : Matrices

Example Question #54 : Matrices

Example Question #61 : Matrices

Example Question #62 : Matrices

Example Question #63 : Matrices

Example Question #64 : Matrices

Example Question #651 : Vectors And Vector Operations

Example Question #62 : Matrices

All Calculus 3 Resources

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