Matrices - AP Calculus BC

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Question

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

Answer

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:

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Question

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

Answer

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:

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Question

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

Answer

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:

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Question

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

Answer

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:

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Question

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

Answer

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:

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Question

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

Answer

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:

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Question

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

Answer

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:

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Question

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

Answer

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:

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Question

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

Answer

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:

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Question

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

Answer

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:

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Question

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

Answer

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 $A=\begin{vmatrix} 1&2 \1 & 5\end{vmatrix}$

The determinant is thus:

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Question

Find the determinant of the matrix $A=\begin{vmatrix} 12&40 \1 &3 \end{vmatrix}$

Answer

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 $A=\begin{vmatrix} 12&40 \1 &3 \end{vmatrix}$

The determinant is thus:

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Question

Find the determinant of the matrix $A=\begin{vmatrix} 4& 2&0 \3 &5 &9 \1 &1 & 6\end{vmatrix}$

Answer

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 $A=\begin{vmatrix} 4& 2&0 \3 &5 &9 \1 &1 & 6\end{vmatrix}$

The determinant is thus:

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Question

Find the determinant of the matrix $A=\begin{vmatrix} 7&0 &0 \2 &1 &1 \ 3&5 &9 \end{vmatrix}$

Answer

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 $A=\begin{vmatrix} 7&0 &0 \2 &1 &1 \ 3&5 &9 \end{vmatrix}$

The determinant is thus:

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Question

Find the determinant of the matrix $A=\begin{vmatrix} 10&6 &14 \3 &8 & 2\11 &1 &9 \end{vmatrix}$

Answer

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 $A=\begin{vmatrix} 10&6 &14 \3 &8 & 2\11 &1 &9 \end{vmatrix}$

The determinant is thus:

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Question

Find the determinant of the 2x2 matrix $\begin{bmatrix} 1&14 \ 2& 5 \end{bmatrix}$

Answer

The formula foe the determinant of a 2x2 matrix $\begin{bmatrix} a&b \ c& d \end{bmatrix}$ is . Using the matrix from the problem statement, we get

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Question

Which of the following is a way to represent the computation $<2,6,1> \cdot <5,2,4>$ using matrices?

Answer

This choice is the only pair with a well-defined operation; the others are meaningless in terms of matrix multiplication.

Computing this we get

$\begin{bmatrix} 2 & 6 & 1 \end{bmatrix} \begin{bmatrix} 5\ 2\ 4\ \end{bmatrix} = (2)(5)+(6)(2)+(1)(4) =26$ .

Which is the same as

$<2,6,1> \cdot <5,2,4> = (2)(5)+(6)(2)+(1)(4) =26$ .

This operation is true for (real) -dimensional vectors in general;

$\begin{bmatrix} a & b & c \end{bmatrix} \begin{bmatrix} d\ e\ f\ \end{bmatrix} = ad+be+cf$

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Question

Find the determinant of the matrix A:

$\mathbf{A}= \begin{pmatrix} 7&4 \ 5&2 \end{pmatrix}$

Answer

The determinant of a matrix

$\begin{pmatrix} a&b \ c&d \end{pmatrix}$

is defined as:

Here, that becomes:

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Question

Calculate the determinant of Matrix .

$A=\begin{bmatrix} 1 & 2&3 \ 0& 2& 1\ 3& 4& 5 \end{bmatrix}$

Answer

In order to find the determinant of , we first need to copy down the first two columns into columns 4 and 5.

$A=\begin{bmatrix} 1 & 2&3 & 1 &2\ 0& 2& 1&0 &2\ 3& 4& 5 &3 &4\end{bmatrix}$

The next step is to multiply the down diagonals.

$A_d=1\cdot2\cdot5+2\cdot1\cdot 3+3\cdot0\cdot4=10+6+0=16$

The next step is to multiply the up diagonals.

$A_u=3\cdot2\cdot3+4\cdot1\cdot1+5\cdot0\cdot2=18+4=22$

The last step is to substract from .

$\det(A)=A_d-A_u=16-22=-6$

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Question

Calculate the determinant of Matrix .

$A=\begin{bmatrix} 0 & 1&5 \ 10& -3& 2\ 1& 2& -5 \end{bmatrix}$

Answer

In order to find the determinant of , we first need to copy down the first two columns into columns 4 and 5.

$A=\begin{bmatrix} 0 & 1&5 & 0 &1\ 10& -3& 2&10 &-3\ 1& 2& -5 &1 &2\end{bmatrix}$

The next step is to multiply the down diagonals.

$A_d=0\cdot-3\cdot-5+1\cdot2\cdot 1+5\cdot10\cdot2=0+2+100=102$

The next step is to multiply the up diagonals.

$A_u=1\cdot-3\cdot5+2\cdot2\cdot0+(-5)\cdot10\cdot1=-15+0-50=-65$

The last step is to substract from .

$\det(A)=A_d-A_u=102-(-65)=167$

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