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Question

A is a square matrix of order n.
l= maximum number of distinct entries if A is a triangular matrix
m= maximum number of distinct entries if A is a diagonal matrix
p= minimum number of zeroes if A is a triangular matrix
If l+5=p+2m, find the order of the matrix.

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Solution

A is a square matrix of order n
L= maximum number of distinct entries if A is triangular.
Lets assume all the non-zero elements in the triangular matrix are distinct.
So maximum number is =1+2+3+.........+n
because in the first column there is one non zero elements, in second column there are two elements and in the nth column n elements.
So here L=1+2+3+.........+nn(n+1)2
m= maximum number of distinct entries if A is diagonal.
Lets assume all the non zero elements are distinct. So in each column there is one non-zero element so m=1+1+1+........+1n numbers
So m=n
P= minimum number of zeros in A if A is triangular
Lets assume all the elements in and above or below of the diagonal in the triangular matrix is non-zero.
So minimum number of 0s=n2n(n+1)2P=2n2n2n2P=n2n2L+5=P+2mn(n+1)2+5=n2n2+2nn2+n+102=n2n+4n2n2+n+10=n2+3n2n=10n=5
So order of the matrix =5

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