In lower triangular matrix of order `nxxn`. Find (i) Minimum number of zeroes. (ii) Maximum number of zeroes.

Let A be a square matrix of order n . l = maximum number of different entries if A is a upper triangular matrix. m = minimum number of zeros if A is a triangular matrix. p = minimum number of zeros if A is a diagonal matrix. If l+2m=2p+1 , then n is :

In a upper triangular matrix ntimesn, minimum number of zeros is

Note:(1) Minimum number of zeros in an upper or lower triangular matrix of ordernn(n-1)= 1+2+3+......... + (n -1)= -2Minimum number of cyphers in a diagonal/scalar/unit matrix of order n=n(n-1)and maximum number of cyphers=n²-1.It is to be noted that with every square matrix there is a corresponding determinant formed by theelements of A in the same order. If|Al=0 then A is called a singular matrix and if|A| =0 then a iscalled a non singular matrix.To 01Note: IfA= o then det. A=0 but not conversely.

If A is a skew symmetric matrix and B is an upper triangular matrix of order 'n', then the ratio of the maximum number of non-zero elements in A to the minimum number of zero elements in B is

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 + 2 m , find the order of the matrix.

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 + 2 m , find the order of the matrix.

If A is an upper triangular matrix of order nxxna n dB is a lower triangular matrix of order nxxna n dB is a lower triangular matrix of order nxxn , then prove that (A^(prime)+B)xx(A+B ') will be a diagonal matrix of order nxxn [assume all elements of A and dB to e non-negative and a element of (A^(prime)+B)xx(A+B^(prime))a sC_(i j) ].