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Calculus 3 : Matrices

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Example Questions

Example Question #685 : Vectors And Vector Operations

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

Possible Answers:

880 $\displaystyle 880$

$\displaystyle -702$

$\displaystyle -660$

1354 $\displaystyle 1354$

$\displaystyle -2104$

Correct answer:

$\displaystyle -660$

Explanation:

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

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

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

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

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

The determinant is thus:

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

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Example Question #686 : Vectors And Vector Operations

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

Possible Answers:

809 $\displaystyle 809$

271 $\displaystyle 271$

$\displaystyle -505$

$\displaystyle -483$

$\displaystyle -1224$

Correct answer:

$\displaystyle -505$

Explanation:

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

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

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

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

The determinant is thus:

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

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Example Question #687 : Vectors And Vector Operations

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

Possible Answers:

$\displaystyle -815$

$\displaystyle -111$

384 $\displaystyle 384$

444 $\displaystyle 444$

$\displaystyle -290$

Correct answer:

$\displaystyle -111$

Explanation:

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

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

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

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

The determinant is thus:

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

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Example Question #688 : Vectors And Vector Operations

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

Possible Answers:

903 $\displaystyle 903$

1544 $\displaystyle 1544$

477 $\displaystyle 477$

$\displaystyle -1113$

$\displaystyle -897$

Correct answer:

$\displaystyle -897$

Explanation:

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

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

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

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

The determinant is thus:

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

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Example Question #689 : Vectors And Vector Operations

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

Possible Answers:

$\displaystyle -288$

$\displaystyle -192$

$\displaystyle 80$

405 $\displaystyle 405$

311 $\displaystyle 311$

Correct answer:

$\displaystyle 80$

Explanation:

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

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

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

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

The determinant is thus:

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

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Example Question #681 : Vectors And Vector Operations

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

Possible Answers:

117 $\displaystyle 117$

$\displaystyle -42$

$\displaystyle -95$

$\displaystyle 13$

$\displaystyle -285$

Correct answer:

117 $\displaystyle 117$

Explanation:

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

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

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

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

The determinant is thus:

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

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Example Question #691 : Vectors And Vector Operations

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

Possible Answers:

972 $\displaystyle 972$

800 $\displaystyle 800$

1551 $\displaystyle 1551$

$\displaystyle 53$

223 $\displaystyle 223$

Correct answer:

800 $\displaystyle 800$

Explanation:

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

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

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

For the matrix $A=\begin{vmatrix} 5&-5 &-3 \\-5 &5 &-7 \\7 &9 &5 \end{vmatrix}$ $\displaystyle A=\begin{vmatrix} 5&-5 &-3 \\-5 &5 &-7 \\7 &9 &5 \end{vmatrix}$

The determinant is thus:

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

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Example Question #692 : Vectors And Vector Operations

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

Possible Answers:

$\displaystyle 96$

$\displaystyle 21$

$\displaystyle 0$

$\displaystyle -255$

$\displaystyle -144$

Correct answer:

$\displaystyle 96$

Explanation:

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

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

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

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

The determinant is thus:

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

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Example Question #693 : Vectors And Vector Operations

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

Possible Answers:

224 $\displaystyle 224$

1004 $\displaystyle 1004$

488 $\displaystyle 488$

125 $\displaystyle 125$

$\displaystyle 33$

Correct answer:

224 $\displaystyle 224$

Explanation:

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

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

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

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

The determinant is thus:

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

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Example Question #694 : Vectors And Vector Operations

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

Possible Answers:

$\displaystyle -1000$

$\displaystyle -401$

$\displaystyle -366$

$\displaystyle -503$

$\displaystyle -11$

Correct answer:

$\displaystyle -401$

Explanation:

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

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

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

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

The determinant is thus:

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

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Calculus 3 : Matrices

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #685 : Vectors And Vector Operations

Example Question #686 : Vectors And Vector Operations

Example Question #687 : Vectors And Vector Operations

Example Question #688 : Vectors And Vector Operations

Example Question #689 : Vectors And Vector Operations

Example Question #681 : Vectors And Vector Operations

Example Question #691 : Vectors And Vector Operations

Example Question #692 : Vectors And Vector Operations

Example Question #693 : Vectors And Vector Operations

Example Question #694 : Vectors And Vector Operations

All Calculus 3 Resources

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