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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 #97 : Matrices

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

Possible Answers:

972

1551

223

800

Correct answer:

800

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 $A=\begin{vmatrix} 5&-5 &-3 \\-5 &5 &-7 \\7 &9 &5 \end{vmatrix}$

The determinant is thus:

|A|=5[5(5)-9(-7)]-(-5)[-5(5)-7(-7)]+(-3)[-5(9)-7(5)]=800

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Example Question #98 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} -6& 0& 2\\8 &-8 &-8 \\8 &1 &0 \end{vmatrix}$

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 $A=\begin{vmatrix} -6& 0& 2\\8 &-8 &-8 \\8 &1 &0 \end{vmatrix}$

The determinant is thus:

|A|=-6[-8(0)-1(-8)]-0[8(0)-8(-8)]+2[8(1)-8(-8)]=96

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Example Question #99 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 8& -2&5 \\ 8& 6& 5\\4 &7 &6 \end{vmatrix}$

Possible Answers:

1004

488

125

224

Correct answer:

224

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 $A=\begin{vmatrix} 8& -2&5 \\ 8& 6& 5\\4 &7 &6 \end{vmatrix}$

The determinant is thus:

|A|=8[6(6)-7(5)]-(-2)[8(6)-4(5)]+5[8(7)-4(6)]=224

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Example Question #100 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 5& 8&-1 \\-2 &7 &-9 \\9 &5 &-1 \end{vmatrix}$

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 $A=\begin{vmatrix} 5& 8&-1 \\-2 &7 &-9 \\9 &5 &-1 \end{vmatrix}$

The determinant is thus:

|A|=5[7(-1)-5(-9)]-8[-2(-1)-9(-9)]+(-1)[-2(5)-9(7)]=-401

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Example Question #101 : Matrices

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

Possible Answers:

$\begin{vmatrix} -12\\-11 \end{vmatrix}$

$\begin{vmatrix} 12\\24 \end{vmatrix}$

$\begin{vmatrix} -8\\-6 \end{vmatrix}$

$\begin{vmatrix} -6\\-8 \end{vmatrix}$

$\begin{vmatrix} -11\\-12 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} -12\\-11 \end{vmatrix}$

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

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

The resulting matrix product is then:

$A\times b = \begin{vmatrix} -16+4\\-12+1 \end{vmatrix}=\begin{vmatrix} -12\\-11 \end{vmatrix}$

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Example Question #691 : Vectors And Vector Operations

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix}-6 & 0\\1 & -2\end{vmatrix}$ and $b=\begin{vmatrix} -1 \\ 0 \end{vmatrix}$

Possible Answers:

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

$\begin{vmatrix} 0\\-1 \end{vmatrix}$

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

$\begin{vmatrix} 6\\-1 \end{vmatrix}$

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

Correct answer:

$\begin{vmatrix} 6\\-1 \end{vmatrix}$

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

$A=\begin{vmatrix}-6 & 0\\1 & -2\end{vmatrix}$ and $b=\begin{vmatrix} -1 \\ 0 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} 6+0\\-1+0 \end{vmatrix}=\begin{vmatrix} 6\\-1 \end{vmatrix}$

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Example Question #103 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 5& 7\\9 &-5 \end{vmatrix}$ and $b=\begin{vmatrix} -2\\-8 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} 45\\-32 \end{vmatrix}$

$\begin{vmatrix} -66\\12 \end{vmatrix}$

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

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

$\begin{vmatrix} -56\\22 \end{vmatrix}$

Correct answer:

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

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

$A=\begin{vmatrix} 5& 7\\9 &-5 \end{vmatrix}$ and $b=\begin{vmatrix} -2\\-8 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}-10-56 \\ -18+40\end{vmatrix}=\begin{vmatrix} -66\\22 \end{vmatrix}$

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Example Question #104 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 4& -9\\6 &9 \end{vmatrix}$ and $b=\begin{vmatrix} 6\\5 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} -21\\81 \end{vmatrix}$

$\begin{vmatrix} -21\\-81 \end{vmatrix}$

$\begin{vmatrix} 21\\81 \end{vmatrix}$

$\begin{vmatrix} 21\\-81 \end{vmatrix}$

$\begin{vmatrix} 7\\-27 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} -21\\81 \end{vmatrix}$

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

$A=\begin{vmatrix} 4& -9\\6 &9 \end{vmatrix}$ and $b=\begin{vmatrix} 6\\5 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} 24-45\\36+45 \end{vmatrix}=\begin{vmatrix} -21\\81 \end{vmatrix}$

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Example Question #105 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} -9&-3 \\2 &6 \end{vmatrix}$ and $b=\begin{vmatrix} 3\\ -8\end{vmatrix}$

Possible Answers:

$\begin{vmatrix} -2\\-16 \end{vmatrix}$

$\begin{vmatrix} -6\\-42 \end{vmatrix}$

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

$\begin{vmatrix} 1\\26 \end{vmatrix}$

$\begin{vmatrix} -3\\-42 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} -3\\-42 \end{vmatrix}$

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

$A=\begin{vmatrix} -9&-3 \\2 &6 \end{vmatrix}$ and $b=\begin{vmatrix} 3\\ -8\end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} -27+24\\6-48 \end{vmatrix}=\begin{vmatrix} -3\\-42 \end{vmatrix}$

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Example Question #106 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} -8& -9\\6 &3 \end{vmatrix}$ and $b=\begin{vmatrix} 7\\-4 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} -20\\30 \end{vmatrix}$

$\begin{vmatrix} 10\\30 \end{vmatrix}$

$\begin{vmatrix} 10\\-30 \end{vmatrix}$

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

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

Correct answer:

$\begin{vmatrix} -20\\30 \end{vmatrix}$

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

$A=\begin{vmatrix} -8& -9\\6 &3 \end{vmatrix}$ and $b=\begin{vmatrix} 7\\-4 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} -56+36\\42-12 \end{vmatrix}=\begin{vmatrix} -20\\30 \end{vmatrix}$

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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 #97 : Matrices

Example Question #98 : Matrices

Example Question #99 : Matrices

Example Question #100 : Matrices

Example Question #101 : Matrices

Example Question #691 : Vectors And Vector Operations

Example Question #103 : Matrices

Example Question #104 : Matrices

Example Question #105 : Matrices

Example Question #106 : Matrices

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