If A = [aij] is a scalar matrix of order n × n such that aii = k, for all i, then trace of A is equal to
Question:
If $A=\left[a_{i j}\right]$ is a scalar matrix of order $n \times n$ such that $a_{i j}=k$, for all $i$, then trace of $A$ is equal to
(a) $n k$
(b) $n+k$
(c) $\frac{n}{k}$
(d) none of these
Solution:
(a) $n k$
Here,
$A=\left[a_{i j}\right]_{n \times n}$
Trace of $A$, i. e. $\operatorname{tr}(A)=\sum_{i=1}^{n} a_{i j}=a_{11}+a_{22}+\ldots+a_{n n}$
$=k+k+k+\ldots n$ times
$=k n$
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