The Trace - Linear Algebra

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Calculate the trace of the following Matrix.

$\begin{bmatrix} 2&4 &45 \ 3& 10& 30\ 100& 3& 56 \end{bmatrix}$

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The trace of a matrix is simply adding the entries along the main diagonal.

Calculate the trace.

$\begin{bmatrix} 1&4 \ 5&-1 \end{bmatrix}$

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The trace of a matrix is simply adding the entries along the main diagonal.

Find the trace of the following matrix.

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

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

Calculate the trace of the following matrix.

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

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

Calculate the trace of matrix , given

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

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By definition,

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

Therefore,

.

Calculate the trace of , or , given

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

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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.

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

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$\begin{align}&$\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 }$A=\begin{bmatrix}-8&-7\-3&0\end{bmatrix}\&(-8)+(0)=-8\end{align}$

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

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$\begin{align}&$\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}$\begin{bmatrix}-10&-7\7&-15\end{bmatrix}\&$\text{We find:}$\&(-10)+(-15)=-25\end{align}$

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

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$\begin{align}&$\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:}$\&(-7)+(5)+(-14)+(1)+(6)+(14)+(-3)+(19)=21\end{align}$

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

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$\begin{align}&$\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 }$A=\begin{bmatrix}20&-8&8&7\2&8&7&-13\-15&20&-13&-19\3&16&7&-13\end{bmatrix}\&Tr(A)=(20)+(8)+(-13)+(-13)=2\end{align}$

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

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$\begin{align}&$\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: }$\&Tr(A)=(-3)+(-11)+(-9)=-23\end{align}$

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

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$\begin{align}&$\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: }$\&(3)+(-10)+(-14)+(-1)=-22\end{align}$

$\begin{align}&$\text{Find }$Tr(A)$\text{ for }$A=\begin{bmatrix}1&6&-4&13\9&19&1&-7\-16&5&11&-3\-17&-10&-14&-9\end{bmatrix}\end{align}$

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$\begin{align}&$\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 }$A=\begin{bmatrix}1&6&-4&13\9&19&1&-7\-16&5&11&-3\-17&-10&-14&-9\end{bmatrix}\&$\text{We find: }$\&Tr(A)=(1)+(19)+(11)+(-9)=22\end{align}$

$\begin{align}&$\text{Find }$Tr(A)$\text{ for }$A=\begin{bmatrix}6&19&-11&7&-9&7&8&-18\-10&-11&7&14&-6&12&7&-20\4&-5&17&-20&-2&-3&-2&11\-7&12&-1&-19&-13&9&-1&-14\-7&4&-13&10&-11&17&-9&11\-13&-9&-17&3&8&2&-3&6\6&7&6&18&-12&9&-11&-16\4&-2&-2&7&11&-6&7&-3\end{bmatrix}\end{align}$

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$\begin{align}&$\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 }$A=\begin{bmatrix}6&19&-11&7&-9&7&8&-18\-10&-11&7&14&-6&12&7&-20\4&-5&17&-20&-2&-3&-2&11\-7&12&-1&-19&-13&9&-1&-14\-7&4&-13&10&-11&17&-9&11\-13&-9&-17&3&8&2&-3&6\6&7&6&18&-12&9&-11&-16\4&-2&-2&7&11&-6&7&-3\end{bmatrix}\&$\text{We find: }$\&Tr(A)=(6)+(-11)+(17)+(-19)+(-11)+(2)+(-11)+(-3)=-30\end{align}$

$\begin{align}&$\text{Calculate the trace of the matrix}$\&\begin{bmatrix}15&-10&-7&-16&18\6&-1&6&2&6\2&9&1&20&-12\-16&-16&-18&-4&-2\-6&11&5&11&18\end{bmatrix}\end{align}$

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$\begin{align}&$\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 }$\begin{bmatrix}15&-10&-7&-16&18\6&-1&6&2&6\2&9&1&20&-12\-16&-16&-18&-4&-2\-6&11&5&11&18\end{bmatrix}\&$\text{This becomes: }$\&(15)+(-1)+(1)+(-4)+(18)=29\end{align}$

$\begin{align}&$\text{Compute }$Tr\begin{bmatrix}1&15&-1&-4&7\10&1&-6&-14&4\-10&-19&10&-11&-2\8&-6&10&-4&8\8&-2&-20&-7&-3\end{bmatrix}\end{align}$

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$\begin{align}&$\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}$\begin{bmatrix}1&15&-1&-4&7\10&1&-6&-14&4\-10&-19&10&-11&-2\8&-6&10&-4&8\8&-2&-20&-7&-3\end{bmatrix}\&$\text{We find:}$\&(1)+(1)+(10)+(-4)+(-3)=5\end{align}$

$\begin{align}&$\text{Find }$Tr(A)$\text{ for }$A=\begin{bmatrix}13&10&-5&-12\12&18&-7&7\-3&14&11&-14\15&20&1&16\end{bmatrix}\end{align}$

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$\begin{align}&$\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 }$A=\begin{bmatrix}13&10&-5&-12\12&18&-7&7\-3&14&11&-14\15&20&1&16\end{bmatrix}\&$\text{We find: }$\&Tr(A)=(13)+(18)+(11)+(16)=58\end{align}$

$\begin{align}&$\text{Find }$Tr(A)$\text{ for }$A=\begin{bmatrix}12&-7&1&-17&-16&-15&7\0&-13&0&-14&-18&14&2\18&8&3&13&16&20&-20\15&5&20&1&-1&12&-11\0&16&3&14&10&4&-10\7&-17&5&7&9&16&20\11&3&18&3&-20&-16&15\end{bmatrix}\end{align}$

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$\begin{align}&$\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 }$A=\begin{bmatrix}12&-7&1&-17&-16&-15&7\0&-13&0&-14&-18&14&2\18&8&3&13&16&20&-20\15&5&20&1&-1&12&-11\0&16&3&14&10&4&-10\7&-17&5&7&9&16&20\11&3&18&3&-20&-16&15\end{bmatrix}\&$\text{We find: }$\&Tr(A)=(12)+(-13)+(3)+(1)+(10)+(16)+(15)=44\end{align}$

$A = \begin{bmatrix} 3&0 & 0\ 0& 6& 0\ 0&0 &x \end{bmatrix}$ ,

where is a complex number. The trace of is 50.

Evaluate .

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is a diagonal matrix - its only nonzero elements are on the main diagonal, from upper left to lower right - so its square can be taken by simply squaring the diagonal elements. Since

$A = \begin{bmatrix} 3&0 & 0\ 0& 6& 0\ 0&0 &x \end{bmatrix}$

it follows that

$A ^{2}= \begin{bmatrix} $3^{2}$&0 & 0\ 0& $6^{2}$& 0\ 0&0 __MATH_EXPR_2x^{2}$ \end{bmatrix} = \begin{bmatrix} 9&0 & 0\ 0& 36 & 0\ 0&0 MATH_EXPR_3__x^{2}$ \end{bmatrix}$

The trace of a matrix is equal to the sum of the elements in its main diagonal, so the trace of is

Since the trace is given to be 50, set this equal to 50 and solve for :

$x= \pm $\sqrt{5}$$

All skew-symmetric matrices have a trace of

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For any skew-symmetric matrix , . Taking the trace of both sides we get

.