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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #11 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 1&4 &2 \\-3 &0 &5 \\2 &7 & 0\end{vmatrix}$

Possible Answers:

108

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 $A=\begin{vmatrix} 1&4 &2 \\-3 &0 &5 \\2 &7 & 0\end{vmatrix}$

The determinant is thus:

|A|=1(0*0-7*5)-4(-3*0-2*5)+2(-3*7-2*0)=-37

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Example Question #12 : Matrices

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

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

The determinant is thus:

|A|=14-8=6

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Example Question #13 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 8& 11\\-4 & 1\end{vmatrix}$

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 $A=\begin{vmatrix} 8& 11\\-4 & 1\end{vmatrix}$

The determinant is thus:

|A|=8-(-44)=52

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Example Question #14 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 3&11 \\18 &2 \end{vmatrix}$

Possible Answers:

204

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 $A=\begin{vmatrix} 3&11 \\18 &2 \end{vmatrix}$

The determinant is thus:

|A|=6-198=-192

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Example Question #21 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} -1&5 &3 \\ 7&11 &2 \\1 &4 &6 \end{vmatrix}$

Possible Answers:

193

147

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 $A=\begin{vmatrix} -1&5 &3 \\ 7&11 &2 \\1 &4 &6 \end{vmatrix}$

The determinant is thus:

|A|=-1(11(6)-4(2))-5(7(6)-1(2))+3(7(4)-1(11))=-207

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Example Question #22 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 13&-5 \\ 3&1 \end{vmatrix}$

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 $A=\begin{vmatrix} 13&-5 \\ 3&1 \end{vmatrix}$

The determinant is thus:

|A|=13-(-15)=28

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Example Question #23 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix}0 &23 \\ 2& 11\end{vmatrix}$

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 $A=\begin{vmatrix}0 &23 \\ 2& 11\end{vmatrix}$

The determinant is thus:

|A|=0-46=-46

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Example Question #21 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 4&20 \\3 &23 \end{vmatrix}$

Possible Answers:

152

149

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 $A=\begin{vmatrix} 4&20 \\3 &23 \end{vmatrix}$

The determinant is thus:

|A|=92-60=32

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Example Question #24 : Matrices

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

Possible Answers:

111

153

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

The determinant is thus:

|A|=4[7(1)-3(8)]-(-1)[2(1)-9(8)]+3[2(3)-9(7)]=-309

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Example Question #22 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 1&2 \\3 &8 \end{vmatrix}$

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 $A=\begin{vmatrix} 1&2 \\3 &8 \end{vmatrix}$

The determinant is thus:

|A|=8-6=2

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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 #11 : Matrices

Example Question #12 : Matrices

Example Question #13 : Matrices

Example Question #14 : Matrices

Example Question #21 : Matrices

Example Question #22 : Matrices

Example Question #23 : Matrices

Example Question #21 : Matrices

Example Question #24 : Matrices

Example Question #22 : Matrices

All Calculus 3 Resources

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