Question

Let \(A\) and \(B\) be any two \(n\times n\) matrices and \[ tr(A)=\sum_{i=1}^{n}a_{ii}, \qquad tr(B)=\sum_{i=1}^{n}b_{ii}. \] Consider the following statements:
• Statement-I : \(tr(AB)=tr(BA)\).
• Statement-II : \(tr(A+B)=tr(A)+tr(B)\).
• Statement-III : If \(tr(A)=5,\; tr(A^{2})=13\), then \[ tr(A-I)^2=6, \] where \(A_{3\times3}\) and \(I_{3\times3}\) are matrices. Choose the correct option.

• A. Statement-I only is true
• B. Statement-I and Statement-II only are true
• C. Statement-I and Statement-III only are true
• D. Statement-I, Statement-II and Statement-III are all true

Hint:

Remember these four trace identities: \[ tr(A+B)=tr(A)+tr(B) \] \[ tr(kA)=k\,tr(A) \] \[ tr(AB)=tr(BA) \] \[ tr(I_n)=n \] Also, \[ tr\!\left((A-I)^2\right) = tr(A^2)-2tr(A)+n, \] where \(n\) is the order of the identity matrix. These formulas are frequently asked in GATE, CUET(PG), NET and Engineering Mathematics examinations.

Answer 1

The Correct Option is D