The Trace

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Linear Algebra › The Trace

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

Explanation

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

and are square matrices of the same dimension.

True or false: $\textup{Tr}(A) + \textup{Tr}(B) = \textup{Tr}(A+B)$

True

False

Explanation

The trace of a square matrix is equal to the sum of its diagonal elements - the elements whose row number and column number are equal. Therefore, if and are $N \times N$ matrices,

$\textup{Tr}(A) = \sum_{j=1}^ {N}a_{jj}$

and

$\textup{Tr}(B) = \sum_{j=1}^ {N}b_{jj}$

By the addition rule for finite sums,

$\textup{Tr}(A)+ \textup{Tr}(B) = \sum_{j=1}^ {N}a_{jj} + \sum_{j=1}^ {N}b_{jj} = \sum_{j=1}^ {N}(a_{jj} + b_{jj})$

Also, addition of matrices is elementwise, so

$\textup{Tr}(A + B) = \sum_{j=1}^ {N}(a_{jj} + b_{jj}) =\textup{Tr}(A)+ \textup{Tr}(B)$

Calculate the trace.

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

Explanation

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

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

Explanation

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

Explanation

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

All skew-symmetric matrices have a trace of

It depends on the skew-symmetric matrix given

Explanation

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

Find the trace of the following matrix.

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

Not possible to calculate

Explanation

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

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

Explanation

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

Explanation

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