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Linear Algebra : Linear Algebra

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Example Questions

Example Question #3 : The Trace

Find the trace of the following matrix.

$\begin{bmatrix} 14 & 15 \\ 2 & -5\\ 5& 10 \end{bmatrix}$

Possible Answers:

Not possible to calculate

Correct answer:

Not possible to calculate

Explanation:

Since the trace can only be calculated for $n\times n$ matrices, the trace isn't possible to calculate.

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Example Question #4 : The Trace

Calculate the trace of the following matrix.

$\begin{bmatrix} -3& 10&11 \\ 3& 25& 20\\ 6& -45& -22 \end{bmatrix}$

Possible Answers:

Trace=18

Trace=48

Trace=-10

Trace=0

Trace=-61

Correct answer:

Trace=0

Explanation:

In order to calculate the trace, we need to sum up each entry along the main diagonal.

Trace=-3+25-22=0

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Example Question #5 : The Trace

Calculate the trace of matrix , given

$A=\begin{bmatrix} 0& 6& 8&22 \\ -13& 0&32 &25 \\ 11& 0&-3 &43 \\ 4&31 &98 &4 \end{bmatrix}$ .

Possible Answers:

tr(A)= 33

tr(A)= 1

tr(A)= 73

tr(A)= -1

Correct answer:

tr(A)= 1

Explanation:

By definition,

$tr(A)= \sum_{i=1}^n a_{ii}$ .

Therefore,

tr(A)= 0+0+(-3)+4=1 .

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Example Question #6 : The Trace

Calculate the trace of , or tr(A) , given

$A= \begin{bmatrix} 1&2 &5 \\ 4&7 &1 \end{bmatrix}$ .

Possible Answers:

tr(A) = 0

tr(A) = 3

The trace does not exist.

tr(A) = 13

tr(A) = 8

Correct answer:

The trace does not exist.

Explanation:

By definition, the trace of a matrix only exists in the matrix is a square matrix. In this case, is not square. Therefore, the trace does not exist.

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Example Question #7 : The Trace

$\begin{align*}&\text{Determine the trace of the matrix}\\&A=\begin{bmatrix}-8&-7\\-3&0\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{We can find the trace of a square nxn matrix by summing its diagonal elements:}\\&\sum_{i=1}^{n}A_{i,i}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{For }A=\begin{bmatrix}-8&-7\\-3&0\end{bmatrix}\\&(-8)+(0)=-8\end{align*}$

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Example Question #8 : The Trace

$\begin{align*}&\text{Compute }Tr\begin{bmatrix}-10&-7\\7&-15\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

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Example Question #9 : The Trace

$\begin{align*}&\text{Compute }Tr\begin{bmatrix}-7&8&-12&-19&10&0&-1&17\\5&5&15&13&3&-13&-11&16\\-19&0&-14&20&9&0&-1&-18\\7&-19&-18&1&-17&13&13&9\\-14&7&1&19&6&12&-2&-3\\13&-17&-15&-13&-4&14&12&-18\\-4&1&-3&6&5&-9&-3&-20\\20&-14&-16&-5&-12&0&-7&19\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Tr in the context of linear algebra represents the trace.}\\&\text{We can find this quantity for a square nxn matrix by summing the elements of its diagonal:}\\&\sum_{i=1}^{n}A_{i,i}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{Doing so we find for}\begin{bmatrix}-7&8&-12&-19&10&0&-1&17\\5&5&15&13&3&-13&-11&16\\-19&0&-14&20&9&0&-1&-18\\7&-19&-18&1&-17&13&13&9\\-14&7&1&19&6&12&-2&-3\\13&-17&-15&-13&-4&14&12&-18\\-4&1&-3&6&5&-9&-3&-20\\20&-14&-16&-5&-12&0&-7&19\end{bmatrix}\\&\text{We find:}\\&(-7)+(5)+(-14)+(1)+(6)+(14)+(-3)+(19)=21\end{align*}$

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Example Question #9 : The Trace

$\begin{align*}&\text{Determine the trace of the matrix}\\&A=\begin{bmatrix}20&-8&8&7\\2&8&7&-13\\-15&20&-13&-19\\3&16&7&-13\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{We can find the trace of a square nxn matrix by summing its diagonal elements:}\\&\sum_{i=1}^{n}A_{i,i}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{For }A=\begin{bmatrix}20&-8&8&7\\2&8&7&-13\\-15&20&-13&-19\\3&16&7&-13\end{bmatrix}\\&Tr(A)=(20)+(8)+(-13)+(-13)=2\end{align*}$

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Example Question #111 : Operations And Properties

$\begin{align*}&\text{Find }Tr(A)\text{ for }A=\begin{bmatrix}-3&-1&-16\\4&-11&-5\\3&-10&-9\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Tr, that is to say the trace, of a square nxn matrix is the its diagonal elements:}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{For the matrix }A=\begin{bmatrix}-3&-1&-16\\4&-11&-5\\3&-10&-9\end{bmatrix}\\&\text{We find: }\\&Tr(A)=(-3)+(-11)+(-9)=-23\end{align*}$

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Example Question #112 : Operations And Properties

$\begin{align*}&\text{Calculate the trace of the matrix}\\&\begin{bmatrix}3&-16&17&16\\13&-10&4&-20\\-3&-8&-14&-13\\-3&-17&4&-1\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{The trace of a square nxn matrix is the sum of its diagonal elements:}\\&\sum_{i=1}^{n}A_{i,i}\\&\text{For the matrix }\begin{bmatrix}3&-16&17&16\\13&-10&4&-20\\-3&-8&-14&-13\\-3&-17&4&-1\end{bmatrix}\\&\text{This becomes: }\\&(3)+(-10)+(-14)+(-1)=-22\end{align*}$

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Linear Algebra : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #3 : The Trace

Example Question #4 : The Trace

Example Question #5 : The Trace

Example Question #6 : The Trace

Example Question #7 : The Trace

Example Question #8 : The Trace

Example Question #9 : The Trace

Example Question #9 : The Trace

Example Question #111 : Operations And Properties

Example Question #112 : Operations And Properties

All Linear Algebra Resources

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