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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