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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 #26 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 1&2 \\1 & 5\end{vmatrix}$

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

The determinant is thus:

|A|=5-2=3

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Example Question #2621 : Calculus 3

Find the determinant of the matrix $A=\begin{vmatrix} 12&40 \\1 &3 \end{vmatrix}$

Possible Answers:

132

477

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 $A=\begin{vmatrix} 12&40 \\1 &3 \end{vmatrix}$

The determinant is thus:

|A|=36-40=-4

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Example Question #25 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 4& 2&0 \\3 &5 &9 \\1 &1 & 6\end{vmatrix}$

Possible Answers:

Correct answer:

Explanation:

The determinant of a 3x3 matrix can be found via a means of reduction into three 2x2 matrices as follows:

$det\begin{vmatrix} a&b&c \\ d&e&f\\g&h&i \end{vmatrix}=a\begin{Vmatrix} e&f \\ h&i \end{Vmatrix}-b\begin{Vmatrix} d&f \\ g&i \end{Vmatrix}+c\begin{Vmatrix} d&e \\ g&h \end{Vmatrix}$

These can then be further reduced via the method of finding the determinant of a 2x2 matrix:

$det\begin{vmatrix} a&b&c \\ d&e&f\\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)$

For the matrix $A=\begin{vmatrix} 4& 2&0 \\3 &5 &9 \\1 &1 & 6\end{vmatrix}$

The determinant is thus:

|A|=4[5(6)-1(9)]-2[3(6)-1(9)]+0[3(1)-1(5)]=66

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Example Question #26 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 7&0 &0 \\2 &1 &1 \\ 3&5 &9 \end{vmatrix}$

Possible Answers:

Correct answer:

Explanation:

The determinant of a 3x3 matrix can be found via a means of reduction into three 2x2 matrices as follows:

$det\begin{vmatrix} a&b&c \\ d&e&f\\g&h&i \end{vmatrix}=a\begin{Vmatrix} e&f \\ h&i \end{Vmatrix}-b\begin{Vmatrix} d&f \\ g&i \end{Vmatrix}+c\begin{Vmatrix} d&e \\ g&h \end{Vmatrix}$

These can then be further reduced via the method of finding the determinant of a 2x2 matrix:

$det\begin{vmatrix} a&b&c \\ d&e&f\\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)$

For the matrix $A=\begin{vmatrix} 7&0 &0 \\2 &1 &1 \\ 3&5 &9 \end{vmatrix}$

The determinant is thus:

|A|=7[1(9)-5(1)]-0[2(9)-3(1)]+0[2(5)-3(1)]=28

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Example Question #621 : Vectors And Vector Operations

Find the determinant of the matrix $A=\begin{vmatrix} 10&6 &14 \\3 &8 & 2\\11 &1 &9 \end{vmatrix}$

Possible Answers:

270

195

138

Correct answer:

Explanation:

The determinant of a 3x3 matrix can be found via a means of reduction into three 2x2 matrices as follows:

$det\begin{vmatrix} a&b&c \\ d&e&f\\g&h&i \end{vmatrix}=a\begin{Vmatrix} e&f \\ h&i \end{Vmatrix}-b\begin{Vmatrix} d&f \\ g&i \end{Vmatrix}+c\begin{Vmatrix} d&e \\ g&h \end{Vmatrix}$

These can then be further reduced via the method of finding the determinant of a 2x2 matrix:

$det\begin{vmatrix} a&b&c \\ d&e&f\\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)$

For the matrix $A=\begin{vmatrix} 10&6 &14 \\3 &8 & 2\\11 &1 &9 \end{vmatrix}$

The determinant is thus:

|A|=10[8(9)-1(2)]-6[3(9)-11(2)]+14[3(1)-11(8)]=-520

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Example Question #622 : Vectors And Vector Operations

Find the determinant of the matrix $A=\begin{vmatrix} 4& 2& 1\\0 &1 &17 \\1 &-1 &13 \end{vmatrix}$

Possible Answers:

279

153

Correct answer:

153

Explanation:

The determinant of a 3x3 matrix can be found via a means of reduction into three 2x2 matrices as follows:

$det\begin{vmatrix} a&b&c \\ d&e&f\\g&h&i \end{vmatrix}=a\begin{Vmatrix} e&f \\ h&i \end{Vmatrix}-b\begin{Vmatrix} d&f \\ g&i \end{Vmatrix}+c\begin{Vmatrix} d&e \\ g&h \end{Vmatrix}$

These can then be further reduced via the method of finding the determinant of a 2x2 matrix:

$det\begin{vmatrix} a&b&c \\ d&e&f\\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)$

For the matrix $A=\begin{vmatrix} 4& 2& 1\\0 &1 &17 \\1 &-1 &13 \end{vmatrix}$

The determinant is thus:

|A|=4[1(13)-(-1)(17)]-2[0(13)-1(17)]+1[0(-1)-1(1)]=153

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Example Question #623 : Vectors And Vector Operations

Find the determinant of the matrix $A=\begin{vmatrix} 10& 13& 85\\3 &23 &50 \\ 4& 20& 29\end{vmatrix}$

Possible Answers:

2799

1341

Correct answer:

Explanation:

The determinant of a 3x3 matrix can be found via a means of reduction into three 2x2 matrices as follows:

$det\begin{vmatrix} a&b&c \\ d&e&f\\g&h&i \end{vmatrix}=a\begin{Vmatrix} e&f \\ h&i \end{Vmatrix}-b\begin{Vmatrix} d&f \\ g&i \end{Vmatrix}+c\begin{Vmatrix} d&e \\ g&h \end{Vmatrix}$

These can then be further reduced via the method of finding the determinant of a 2x2 matrix:

$det\begin{vmatrix} a&b&c \\ d&e&f\\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)$

For the matrix $A=\begin{vmatrix} 10& 13& 85\\3 &23 &50 \\ 4& 20& 29\end{vmatrix}$

The determinant is thus:

|A|=10[23(29)-20(50)]-13[3(29)-4(50)]+85[3(20)-4(23)]=-4581

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Example Question #32 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 2& 5\\ 1& 6\end{vmatrix}$ and $b=\begin{vmatrix} 4\\20 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} 28\\140 \end{vmatrix}$

$\begin{vmatrix} 140\\28 \end{vmatrix}$

$\begin{vmatrix} 108\\124 \end{vmatrix}$

$\begin{vmatrix} 124\\108 \end{vmatrix}$

$\begin{vmatrix} 70\\14 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} 108\\124 \end{vmatrix}$

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 $A=\begin{vmatrix} 2& 5\\ 1& 6\end{vmatrix}$ and $b=\begin{vmatrix} 4\\20 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} 8+100\\4+120 \end{vmatrix}=\begin{vmatrix} 108\\124 \end{vmatrix}$

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Example Question #35 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 5& 0\\3 &9 \end{vmatrix}$ and $b=\begin{vmatrix} 1\\2 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} 24\\5 \end{vmatrix}$

$\begin{vmatrix} 11\\18 \end{vmatrix}$

$\begin{vmatrix} 5\\22 \end{vmatrix}$

$\begin{vmatrix} 5\\24 \end{vmatrix}$

$\begin{vmatrix} 18\\11 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} 5\\22 \end{vmatrix}$

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 $A=\begin{vmatrix} 5& 0\\3 &9 \end{vmatrix}$ and $b=\begin{vmatrix} 1\\2 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}5+0 \\3+19 \end{vmatrix}=\begin{vmatrix} 5\\22 \end{vmatrix}$

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Example Question #36 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 5& 7\\-3 &4 \end{vmatrix}$ and $b=\begin{vmatrix}0 \\ 6\end{vmatrix}$

Possible Answers:

$\begin{vmatrix} 42\\24 \end{vmatrix}$

$\begin{vmatrix} 0\\6 \end{vmatrix}$

$\begin{vmatrix} 24\\-18\end{vmatrix}$

$\begin{vmatrix} -18\\24 \end{vmatrix}$

$\begin{vmatrix} 24\\42 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} 42\\24 \end{vmatrix}$

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 $A=\begin{vmatrix} 5& 7\\-3 &4 \end{vmatrix}$ and $b=\begin{vmatrix}0 \\ 6\end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} 0+42\\0+24 \end{vmatrix}=\begin{vmatrix} 42\\24 \end{vmatrix}$

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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 #26 : Matrices

Example Question #2621 : Calculus 3

Example Question #25 : Matrices

Example Question #26 : Matrices

Example Question #621 : Vectors And Vector Operations

Example Question #622 : Vectors And Vector Operations

Example Question #623 : Vectors And Vector Operations

Example Question #32 : Matrices

Example Question #35 : Matrices

Example Question #36 : Matrices

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