First slide

Operations of matrix

Question

$The trace Tr (A) of a 3 \times 3 matrix A = (a_{i j}) is defined by the$ $relation Tr (A) = a_{11} + a_{22} + a_{33} (i.e., T r (A) is sum of the main$ $diagonal elements). Which of the following statements$ $cannot hold '?$

Easy

A

$T r (kA) = kT r (A) (k is a scalar)$
B

$T r (A + B) = T r (A) + T r (B)$
C

$T r (I_{3}) = 3$
D

$T r (A^{2}) = {[T r (A)]}^{2}$

Solution

$(a) Tr (kA) = k (a_{11} + a_{22} + a_{33}) = kT r (A)$
$(b) Tr (A + B) = a_{11} + b_{11} + a_{22} + b_{22} + a_{33} + b_{33} = Tr (A) + Tr (B)$
$(c) Tr (I_{3}) = 1 + 1 + 1 = 3$
$(d) Tr (A^{2}) = Σ a_{11}^{2} + Σ a_{12}^{2} \neq {(a_{11} + a_{22} + a_{33})}^{2} = {[T r (A)]}^{2}$

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