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Calculus 3 : Matrices

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Example Questions

Example Question #41 : Matrices

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 #42 : Matrices

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 #43 : Matrices

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 #44 : Matrices

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 #45 : Matrices

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 #46 : Matrices

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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Example Question #47 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 7& 11\\3 &23 \end{vmatrix}$

Possible Answers:

128

161

194

275

Correct answer:

128

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 $A=\begin{vmatrix} 7& 11\\3 &23 \end{vmatrix}$

The determinant is thus:

|A|=161-33=128

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Example Question #48 : Matrices

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

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

The determinant is thus:

|A|=8-3=5

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Example Question #49 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix}12 &4 \\30 &10 \end{vmatrix}$

Possible Answers:

400

320

348

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 $A=\begin{vmatrix}12 &4 \\30 &10 \end{vmatrix}$

The determinant is thus:

|A|=120-120=0

This result is due to the columns being linearly dependent, i.e. multiples of each other. The first column is three times the second column.

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Example Question #50 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 1& 3\\7 &13 \end{vmatrix}$

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 $A=\begin{vmatrix} 1& 3\\7 &13 \end{vmatrix}$

The determinant is thus:

|A|=13-21=-8

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Calculus 3 : Matrices

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #41 : Matrices

Example Question #42 : Matrices

Example Question #43 : Matrices

Example Question #44 : Matrices

Example Question #45 : Matrices

Example Question #46 : Matrices

Example Question #47 : Matrices

Example Question #48 : Matrices

Example Question #49 : Matrices

Example Question #50 : Matrices

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