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Question

(A) An upper triangular matrix $A = [a_{ij}]$ is a square matrix where all elements below the main diagonal are zero,i.e.,$a_{ij} = 0$ for all $i > j$

Vedclass · Accepted Answer

$n(n - 1)/2$

Vedclass · Answer

(A) An upper triangular matrix $A = [a_{ij}]$ is a square matrix where all elements below the main diagonal are zero,i.e.,$a_{ij} = 0$ for all $i > j$.For an $n 	imes n$ matrix,the number of elements below the main diagonal is given by the sum of the first $(n-1)$ integers:$1 + 2 + 3 + ... + (n - 1) = \frac{(n - 1)n}{2} = \frac{n(n - 1)}{2}$.For example,in a $4 	imes 4$ matrix,the number of zeros is $\frac{4(4 - 1)}{2} = \frac{12}{2} = 6$.Thus,the minimum number of zeros is $\frac{n(n - 1)}{2}$.

In an upper triangular matrix of order $n \times n$,the minimum number of zeros is:

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