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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #33 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 4&3 &-3 \\1 &6 &2 \\-2 &0 &5 \end{vmatrix}$ and $b=\begin{vmatrix} 0\\ 3\\2 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} 22\\3 \\10 \end{vmatrix}$

$\begin{vmatrix} 10\\3 \\22 \end{vmatrix}$

$\begin{vmatrix} 3\\22 \\10 \end{vmatrix}$

$\begin{vmatrix} 3\\10 \\22 \end{vmatrix}$

$\begin{vmatrix} 22\\10 \\3 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} 3\\22 \\10 \end{vmatrix}$

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 $A=\begin{vmatrix} 4&3 &-3 \\1 &6 &2 \\-2 &0 &5 \end{vmatrix}$ and $b=\begin{vmatrix} 0\\ 3\\2 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} 0+9-6\\0+18+4 \\0+0+10 \end{vmatrix}=\begin{vmatrix} 3\\22 \\10 \end{vmatrix}$

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Example Question #34 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} -4& -1& 8\\2 &3 &5 \\1 &4 &-1 \end{vmatrix}$ and $b=\begin{vmatrix} 3\\2 \\1 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} 10\\17 \\ -6\end{vmatrix}$

$\begin{vmatrix} 17\\10 \\ -6\end{vmatrix}$

$\begin{vmatrix} 10\\-6 \\ 17\end{vmatrix}$

$\begin{vmatrix} -6\\17 \\ 10\end{vmatrix}$

$\begin{vmatrix} 17\\-6 \\ 10\end{vmatrix}$

Correct answer:

$\begin{vmatrix} -6\\17 \\ 10\end{vmatrix}$

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 $A=\begin{vmatrix} -4& -1& 8\\2 &3 &5 \\1 &4 &-1 \end{vmatrix}$ and $b=\begin{vmatrix} 3\\2 \\1 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} -12-2+8\\6+6+5 \\3+8-1 \end{vmatrix}=\begin{vmatrix} -6\\17 \\ 10\end{vmatrix}$

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Example Question #39 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 1& 6& 14\\1 &1 &3 \\2 &1 &5 \end{vmatrix}$ and $b=\begin{vmatrix} 2\\0 \\6 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} 34\\ 20\\86 \end{vmatrix}$

$\begin{vmatrix} 20\\ 34\\86 \end{vmatrix}$

$\begin{vmatrix} 86\\ 20\\34 \end{vmatrix}$

$\begin{vmatrix} 20\\ 86\\34 \end{vmatrix}$

$\begin{vmatrix} 86\\ 34\\20 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} 86\\ 20\\34 \end{vmatrix}$

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 $A=\begin{vmatrix} 1& 6& 14\\1 &1 &3 \\2 &1 &5 \end{vmatrix}$ and $b=\begin{vmatrix} 2\\0 \\6 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}2+0+84 \\2+0+18 \\4+0+30 \end{vmatrix}=\begin{vmatrix} 86\\ 20\\34 \end{vmatrix}$

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Example Question #631 : Vectors And Vector Operations

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

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

The determinant is thus:

|A|=1[2(1)-2(3)]-1[1(1)-3(3)]+1[1(2)-3(2)]=0

(It is of note that if a matrix has a zero determinant, then its columns are linearly dependent. In other words, one column could be created by some via some combination of the other two.

Note how if you multiply the second column by two and then subtract the first column, the third column results.)

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Example Question #632 : Vectors And Vector Operations

Find the determinant of the matrix $A=\begin{vmatrix}6 &4 &2 \\11 & -8& 2\\3 &-9 &4 \end{vmatrix}$

Possible Answers:

132

504

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 $A=\begin{vmatrix}6 &4 &2 \\11 & -8& 2\\3 &-9 &4 \end{vmatrix}$

The determinant is thus:

|A|=6[-8(4)-(-9)(2)]-4[11(4)-3(2)]+2[11(-9)-3(-8)]=-386

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Example Question #633 : Vectors And Vector Operations

Find the determinant of the matrix $A=\begin{vmatrix} 9&-1 &10 \\2 &3 &1 \\4 &8 &7 \end{vmatrix}$

Possible Answers:

167

Correct answer:

167

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 $A=\begin{vmatrix} 9&-1 &10 \\2 &3 &1 \\4 &8 &7 \end{vmatrix}$

The determinant is thus:

|A|=9[3(7)-8(1)]-(-1)[2(7)-4(1)]+10[2(8)-4(3)]=167

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Example Question #634 : Vectors And Vector Operations

Find the determinant of the matrix $A=\begin{vmatrix} 2&3 &5 \\7 &11 &13 \\17 &19 &23 \end{vmatrix}$

Possible Answers:

102

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 $A=\begin{vmatrix} 2&3 &5 \\7 &11 &13 \\17 &19 &23 \end{vmatrix}$

The determinant is thus:

|A|=2[11(23)-19(13)]-3[7(23)-17(13)]+5[7(19)-17(11)]=-78

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Example Question #635 : Vectors And Vector Operations

Find the determinant of the matrix $A=\begin{vmatrix} 83& 79& 73\\71 &67 &61 \\ 53&47 &43 \end{vmatrix}$

Possible Answers:

402

1013

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 $A=\begin{vmatrix} 83& 79& 73\\71 &67 &61 \\ 53&47 &43 \end{vmatrix}$

The determinant is thus:

|A|=83[67(43)-47(61)]-79[71(43)-53(61)]+73[71(47)-53(67)]=-240

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Example Question #636 : Vectors And Vector Operations

Find the determinant of the matrix $A=\begin{vmatrix} 3&13 &15 \\ 20& -4&7 \\2 &11 &14 \end{vmatrix}$

Possible Answers:

803

6479

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 $A=\begin{vmatrix} 3&13 &15 \\ 20& -4&7 \\2 &11 &14 \end{vmatrix}$

The determinant is thus:

|A|=3[-4(14)-11(7)]-13[20(14)-2(7)]+15[20(11)-2(-4)]=-437

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Example Question #637 : Vectors And Vector Operations

Find the determinant of the matrix $A=\begin{vmatrix} 4&3 &4 \\2 &0 &6 \\0 &8 &3 \end{vmatrix}$

Possible Answers:

404

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 $A=\begin{vmatrix} 4&3 &4 \\2 &0 &6 \\0 &8 &3 \end{vmatrix}$

The determinant is thus:

|A|=4[0(3)-8(6)]-3[2(3)-0(6)]+4[2(8)-0(0)]=-146

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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 #33 : Matrices

Example Question #34 : Matrices

Example Question #39 : Matrices

Example Question #631 : Vectors And Vector Operations

Example Question #632 : Vectors And Vector Operations

Example Question #633 : Vectors And Vector Operations

Example Question #634 : Vectors And Vector Operations

Example Question #635 : Vectors And Vector Operations

Example Question #636 : Vectors And Vector Operations

Example Question #637 : Vectors And Vector Operations

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