If a, b, c and are real numbers and A `=[{:( a,b),(c,d) :}]` prove that ` A^(2) -(a+d) A+(ad-bc) I=0`

If A=[(a,b),(c,d)] , where a, b, c and d are real numbers, then prove that A^(2)-(a+d)A+(ad-bc) I=O . Hence or therwise, prove that if A^(3)=O then A^(2)=O

If a, b, c and d are positive real numbers such that a+b+c+d= 1 then prove that ab + bc + cd + da le 1/4 .

If a, b, c, d are positive numbers in HP then prove that ad gt bc .

Statement 1: if a ,b ,c ,d are real numbers and A=[a b c d]a n dA^3=O ,t h e nA^2=Odot Statement 2: For matrix A=[a b c d] we have A^2=(a+d)A+(a d-b c)I=Odot

If a, b, c, d are real numbers such that (a+2c)/(b+3d)+(4)/(3)=0 . Prove that the equation ax^(3)+bx^(2)+cx+d=0 has atleast one real root in (0, 1).

if a,b, c, d and p are distinct real number such that (a^(2) + b^(2) + c^(2))p^(2) - 2p (ab + bc + cd) + (b^(2) + c^(2) + d^(2)) lt 0 then a, b, c, d are in

If {:A=[(a,b),(c,d)]:} such that ad - bc ne 0 , then A^(-1) , is

If a, b, c and d are four positive real numbers such that sum of a, b and c is even the sum of b,c and d is odd, then a^2-d^2 is necessarily