Trace of a Matrix
Category : JEE Main & Advanced
The sum of diagonal elements of a square matrix. A is called the trace of matrix A, which is denoted by tr A.
\[tr\,\,A=\sum\limits_{i=1}^{n}{{{a}_{ii}}={{a}_{11}}+{{a}_{22}}+...{{a}_{nn}}}\]
Properties of trace of a matrix
Let \[{{C}_{11}},\,{{C}_{12}},\,{{C}_{13}}\]and \[B={{[{{b}_{ij}}]}_{n\times n}}\]and \[\lambda \]be a scalar
(i) \[tr(\lambda A)=\lambda \,tr(A)\]
(ii) \[tr(A-B)=tr(A)-\,tr\,(B)\]
(iii) \[tr(AB)=tr(BA)\]
(iv) \[tr\,(A)\,=tr\,(A')\] or \[t{{r}_{{}}}({{A}^{T}})\]
(v) \[tr\,({{I}_{n}})=n\]
(vi) \[tr\,(0)\,=0\]
(vii) \[tr\,(AB)\ne tr\,A\,.\,tr\,B\]
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