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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #81 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} -12 & -6 \\ 4 & 10 \end{vmatrix}$

Possible Answers:

144

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 $A=\begin{vmatrix} -12 & -6 \\ 4 & 10 \end{vmatrix}$

The determinant is thus:

|A|=-120-(-24)=-96

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Example Question #84 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix}7& -15\\-12 &3 \end{vmatrix}$

Possible Answers:

159

201

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 $A=\begin{vmatrix}7& -15\\-12 &3 \end{vmatrix}$

The determinant is thus:

|A|=21-180=-159

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Example Question #89 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} -1&-12 &12 \\15 &11 &3 \\14 &2 &-8 \end{vmatrix}$

Possible Answers:

4113

901

1597

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 $A=\begin{vmatrix} -1&-12 &12 \\15 &11 &3 \\14 &2 &-8 \end{vmatrix}$

The determinant is thus:

{|A|=-1[11(-8)-2(3)]-(-12)[15(-8)-14(3)]+12[15(2)-14(11)]=-3338}

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Example Question #90 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 10&-15 &9 \\-12 & -9& 0\\-4 & 3& 12\end{vmatrix}$

Possible Answers:

5995

2042

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 $A=\begin{vmatrix} 10&-15 &9 \\-12 & -9& 0\\-4 & 3& 12\end{vmatrix}$

The determinant is thus:

{|A|=10[-9(12)-3(0)]-(-15)[-12(12)-(-4)(0)]+9[-12(3)-(-4)(-9)]=-3888}

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Example Question #91 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 15& -15& -15\\10 &-13 &-9 \\11 &3 &-1 \end{vmatrix}$

Possible Answers:

880

1354

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 $A=\begin{vmatrix} 15& -15& -15\\10 &-13 &-9 \\11 &3 &-1 \end{vmatrix}$

The determinant is thus:

{|A|=15[-13(-1)-3(-9)]-(-15)[10(-1)-11(-9)]+(-15)[10(3)-11(-13)]=-660}

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Example Question #92 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 5& 0&-5 \\-7 &8 &-6 \\ -2&-9 &4 \end{vmatrix}$

Possible Answers:

809

271

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 $A=\begin{vmatrix} 5& 0&-5 \\-7 &8 &-6 \\ -2&-9 &4 \end{vmatrix}$

The determinant is thus:

{|A|=5[8(4)-(-9)(-6)]-0[-7(4)-(-2)(-6)]+(-5)[-7(-9)-(-2)(8)]=-505}

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Example Question #93 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix}-6 &-9 &-5 \\-1 &8 &6 \\-3 &6 & 7\end{vmatrix}$

Possible Answers:

444

384

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 $A=\begin{vmatrix}-6 &-9 &-5 \\-1 &8 &6 \\-3 &6 & 7\end{vmatrix}$

The determinant is thus:

{|A|=-6[8(7)-6(6)]-(-9)[-1(7)-(-3)(6)]+(-5)[-1(6)-(-3)(8)]=-111}

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Example Question #94 : Matrices

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

Possible Answers:

477

903

1544

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

The determinant is thus:

|A|=9[-5(8)-9(7)]-8[2(8)-8(7)]+(-5)[2(9)-8(-5)]=-897

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Example Question #95 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 2& -6&8 \\4 &-4 &2 \\8 &0 & -5\end{vmatrix}$

Possible Answers:

405

311

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 $A=\begin{vmatrix} 2& -6&8 \\4 &-4 &2 \\8 &0 & -5\end{vmatrix}$

The determinant is thus:

|A|=2[-4(-5)-0(2)]-(-6)[4(-5)-8(2)]+8[4(0)-8(-4)]=80

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Example Question #96 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} -5&5 &-6 \\3 &-2 &8 \\2 &3 &1 \end{vmatrix}$

Possible Answers:

117

Correct answer:

117

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 $A=\begin{vmatrix} -5&5 &-6 \\3 &-2 &8 \\2 &3 &1 \end{vmatrix}$

The determinant is thus:

|A|=-5[-2(1)-3(8)]-5[3(1)-2(8)]+(-6)[3(3)-2(-2)]=117

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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 #81 : Matrices

Example Question #84 : Matrices

Example Question #89 : Matrices

Example Question #90 : Matrices

Example Question #91 : Matrices

Example Question #92 : Matrices

Example Question #93 : Matrices

Example Question #94 : Matrices

Example Question #95 : Matrices

Example Question #96 : Matrices

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