A is a square matrix of order n. l=maximum number of distinct enteries if A is a triangular matrix. m=maximum number of distinct enteries 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.

Given: A is a square matrix of order n

then

l = n + (n – 1) + (n – 2) ...... (n – (n – 1))

m = n

p = (n – 1) + (n – 2) ...... (n – (n – 1))

also

l + 5 = p + 2m

⇒ n + (n – 1) + (n – 2) ...... (n – (n – 1)) + 5 = (n – 1) + (n – 2) ...... (n – (n – 1)) + 2n

⇒ n + 5 = 2n

⇒ 2n – n = 5

⇒ n = 5

Hence the required order of matrix A is 5.