# JEE Main & Advanced Mathematics Matrices Trace of a Matrix

## 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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