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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #43 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 7& 11\\3 &23 \end{vmatrix}$

Possible Answers:

194

161

128

275

Correct answer:

128

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 $A=\begin{vmatrix} 7& 11\\3 &23 \end{vmatrix}$

The determinant is thus:

|A|=161-33=128

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Example Question #44 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} -4& -3\\-1 & -2\end{vmatrix}$

Possible Answers:

Correct answer:

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 $A=\begin{vmatrix} -4& -3\\-1 & -2\end{vmatrix}$

The determinant is thus:

|A|=8-3=5

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Example Question #45 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix}12 &4 \\30 &10 \end{vmatrix}$

Possible Answers:

320

400

348

Correct answer:

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 $A=\begin{vmatrix}12 &4 \\30 &10 \end{vmatrix}$

The determinant is thus:

|A|=120-120=0

This result is due to the columns being linearly dependent, i.e. multiples of each other. The first column is three times the second column.

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Example Question #46 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 1& 3\\7 &13 \end{vmatrix}$

Possible Answers:

Correct answer:

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 $A=\begin{vmatrix} 1& 3\\7 &13 \end{vmatrix}$

The determinant is thus:

|A|=13-21=-8

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Example Question #641 : Vectors And Vector Operations

Find the determinant of the matrix $A=\begin{vmatrix} -15& -2\\ 4& -1\end{vmatrix}$

Possible Answers:

Correct answer:

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 $A=\begin{vmatrix} -15& -2\\ 4& -1\end{vmatrix}$

The determinant is thus:

|A|=15-(-8)=23

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Example Question #641 : Vectors And Vector Operations

Find the determinant of the matrix $A=\begin{vmatrix} -8&7 \\2 & -3\end{vmatrix}$

Possible Answers:

Correct answer:

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 $A=\begin{vmatrix} -8&7 \\2 & -3\end{vmatrix}$

The determinant is thus:

|A|=24-14=10

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Example Question #643 : Vectors And Vector Operations

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

Possible Answers:

$\begin{vmatrix}10\\8 \end{vmatrix}$

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

$\begin{vmatrix}8\\10 \end{vmatrix}$

$\begin{vmatrix}0\\10 \end{vmatrix}$

$\begin{vmatrix}8\\0 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}8\\10 \end{vmatrix}$

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 $A=\begin{vmatrix} 2&0 \\3 &1 \end{vmatrix}$ and $b=\begin{vmatrix} 4\\-2 \end{vmatrix}$

The resulting matrix product is then:

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

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Example Question #644 : Vectors And Vector Operations

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

Possible Answers:

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

$\begin{vmatrix}12\\4 \end{vmatrix}$

$\begin{vmatrix}0\\12 \end{vmatrix}$

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

$\begin{vmatrix}3\\12 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}3\\12 \end{vmatrix}$

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 $A=\begin{vmatrix} 3&0 \\0 &3 \end{vmatrix}$ and $b=\begin{vmatrix} 1\\4 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} 3+0\\0+12 \end{vmatrix}=\begin{vmatrix}3\\12 \end{vmatrix}$

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Example Question #645 : Vectors And Vector Operations

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 2&3 \\ 4&5 \end{vmatrix}$ and $b=\begin{vmatrix} 6\\7 \end{vmatrix}$ .

Possible Answers:

$\begin{vmatrix}33\\59 \end{vmatrix}$

$\begin{vmatrix}49\\17 \end{vmatrix}$

$\begin{vmatrix}13\\12 \end{vmatrix}$

$\begin{vmatrix}17\\48 \end{vmatrix}$

$\begin{vmatrix}17\\49 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}33\\59 \end{vmatrix}$

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 $A=\begin{vmatrix} 2&3 \\ 4&5 \end{vmatrix}$ and $b=\begin{vmatrix} 6\\7 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}12+21\\24+35 \end{vmatrix}=\begin{vmatrix}33\\59 \end{vmatrix}$

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Example Question #646 : Vectors And Vector Operations

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix}2 & 1\\ 0&9 \end{vmatrix}$ and $b=\begin{vmatrix} 3\\ -2\end{vmatrix}$ .

Possible Answers:

$\begin{vmatrix}8\\18 \end{vmatrix}$

$\begin{vmatrix}-18\\4 \end{vmatrix}$

$\begin{vmatrix}4\\18 \end{vmatrix}$

$\begin{vmatrix}18\\8 \end{vmatrix}$

$\begin{vmatrix}4\\-18 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}4\\-18 \end{vmatrix}$

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 $A=\begin{vmatrix}2 & 1\\ 0&9 \end{vmatrix}$ and $b=\begin{vmatrix} 3\\ -2\end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}6-2\\0-18 \end{vmatrix}=\begin{vmatrix}4\\-18 \end{vmatrix}$

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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 #43 : Matrices

Example Question #44 : Matrices

Example Question #45 : Matrices

Example Question #46 : Matrices

Example Question #641 : Vectors And Vector Operations

Example Question #641 : Vectors And Vector Operations

Example Question #643 : Vectors And Vector Operations

Example Question #644 : Vectors And Vector Operations

Example Question #645 : Vectors And Vector Operations

Example Question #646 : Vectors And Vector Operations

All Calculus 3 Resources

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