Trace of a matrix and properties - Practice Questions & MCQ
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9 Questions around this concept.
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MEDIUMLet $R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$
be a non-zero $3 \times 3$ matrix, where $x \sin \theta=$ $y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0, \theta \in(0,2 \pi)$
For a square matrix M, let trace (M) denote the sum of all the diagonal entries of M. Then, among the statements:
(I) Trace (R) = 0
(II) If three (adj(adj(R)) = 0, then R has exactly one non–zero entry.
If A is $2 \times 2$ matrix (non-zero matrix) such that $A^2=0$, where 0 is a null matrix. Then $\operatorname{tr}(\mathrm{A})=$ ?
If the element of a matrix A is defined by $a_{ij}= i^2-j^2$ and A is a square matrix of order 3 x 3. Then $tr(A)=$
Concepts Covered - 1
Trace of the matrix:
The sum of all diagonal elements of a square matrix is called the trace of a matrix.
The trace of the matrix is denoted by $\operatorname{Tr}(\mathrm{A})$.
$
\operatorname{Tr}(\mathrm{A})=\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{a}_{\mathrm{ii}}
$
For a given matrix A,
$
A=\left[\begin{array}{ccc}
-2 & 4 & 7 \\
8 & 3 & -1 \\
5 & -6 & 9
\end{array}\right], \quad \operatorname{Tr}(\mathrm{A})=-2+3+9=10
$
Properties of a trace of the matrix:
Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{n} \times \mathrm{n}} ; \mathrm{B}=\left[\mathrm{b}_{\mathrm{ij}}\right]_{\mathrm{n} \times \mathrm{n} \text { and } \mathrm{k} \text { is a scalar, then }}$
i) $\operatorname{Tr}(\mathrm{kA})=\mathrm{k} \cdot \operatorname{Tr}(\mathrm{A})$
ii) $\operatorname{Tr}(\mathrm{A} \pm \mathrm{B})=\operatorname{Tr}(\mathrm{A}) \pm \operatorname{Tr}(\mathrm{B})$
iii) $\operatorname{Tr}(\mathrm{AB})=\operatorname{Tr}(B A)$
iv) $\operatorname{Tr}(\mathrm{A})=\operatorname{Tr}\left(\mathrm{A}^{\prime}\right)$
v) $\operatorname{Tr}(A B) \neq \operatorname{Tr}(A) \cdot \operatorname{Tr}(B)$
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