Past exam of the numerical linear algebra course of the University of Oxford / 2011 / 3 / d / Solution

Since $A{p}_1=(5,2{)}^T$, an A-conjugate direction $p_2=(a,b{)}^T$ must satisfy $5a+2b=0$. One nonzero choice isDirect matrix multiplication gives $A{p}_2=(3,-3{)}^T$, and $p_1^{T}A{p}_2=0$. All nonzero scalar multiples of this vector also qualify.