Question

If $ Tr\left( A \right) = 8,\;Tr\left( B \right) = 6 $ then find $ Tr\left( {A - 2B} \right) $ .
(A) $ - 4 $
(B) $ 4 $
(C) $ 2 $
(D) $ 11 $

Answer 1

Hint: To solve the given problem, we will use the following properties of trace of a matrix:
(i) The trace of a sum of two matrices is equal to the sum of their trace. That is, $ Tr\left( {A + B} \right) = Tr\left( A \right) + Tr\left( B \right) $ .
(ii) For any scalar $ \alpha $ and any $ n \times n $ matrix $ A $ , we can write $ Tr\left( {\alpha A} \right) = \alpha Tr\left( A \right) $ .

Complete step-by-step answer:
In this problem, it is given that $ Tr\left( A \right) = 8,\;Tr\left( B \right) = 6 $ . To find $ Tr\left( {A - 2B} \right) $ , let us use the property of trace of matrix which is given by $ Tr\left( {A + B} \right) = Tr\left( A \right) + Tr\left( B \right) $ . So, we can write
 $ Tr\left( {A - 2B} \right) = Tr\left( A \right) + Tr\left( { - 2B} \right) \cdots \cdots \left( 1 \right) $
Now we are going to use the another property of trace of matrix in second term of RHS of equation $ \left( 1 \right) $ and that property is given by
$ Tr\left( {\alpha A} \right) = \alpha Tr\left( A \right) $ where $ \alpha $ is any scalar and $ A $ is square matrix. So, from equation $ \left( 1 \right) $ we can write
 $\Rightarrow Tr\left( {A - 2B} \right) = Tr\left( A \right) - 2Tr\left( B \right) \cdots \cdots \left( 2 \right) $
Now we are going to substitute the given values $ Tr\left( A \right) = 8,\;Tr\left( B \right) = 6 $ in equation $ \left( 2 \right) $ . So, we get
$\Rightarrow Tr\left( {A - 2B} \right) = 8 - 2\left( 6 \right) = 8 - 12 = - 4 $
Hence, we can say that if $ Tr\left( A \right) = 8,\;Tr\left( B \right) = 6 $ then $ Tr\left( {A - 2B} \right) $ is equal to $ - 4 $ .
So, the correct answer is “Option A”.

Note: For any square matrix $ A $ , we can say that the trace of matrix $ A $ is denoted by $ Tr\left( A \right) $ and it can be obtained by finding the sum of principal (main) diagonal elements. If we have $ A = \left[ {\begin{array}{*{20}{c}}
  1&2&3 \\
  4&5&6 \\
  7&8&9
\end{array}} \right] $ then the trace of the matrix $ A $ is equal to $ 1 + 5 + 9 = 15 $ . In this type of problem, we must remember the properties of the trace of the matrix. One more property of the trace of the matrix is given by $ Tr\left( A \right) = Tr\left( {{A^T}} \right) $ where $ {A^T} $ is the transpose of the matrix $ A $ .