JEE Main & Advanced Mathematics Determinants & 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.




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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