Determinantal approach it is known that an R-dimensional square matrix with all elements equal to unity has roots R and 0, with the R occurring once and the zero occurring R – 1 times. If all elements have the value p, then the roots are R_p arid 0. 

(a) Show that if the diagonal elements are q and all other elements are p. then there is one root equal to (R – 1)p + q and R – 1 roots equal to q – p. 

(b) Show from the elastic equation (57) for a wave in the [111] direction of a cubic crystal that the determinantal equation which gives w² as a function of K is where q = 1/3K² (C₁₁ + 2C₄₄) and p = 1/3K²(C₁₂ + C₄₄). This expresses the condition that three linear homogeneous algebraic equations for the three displacement components u, v, w have a solution. Use the result of part (a) to find the three roots of w₂; check with the results given for Problems 9 and 10.