4.(a) Show that the 2 × 2 matrix A commutes with matrix B, where B=\left[\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right] if and only if A is diagonal

\text { (b) Let } B=\left[\begin{array}{ccc} 1-x & 1 & -1 \\ 2 & 1-x & 2 \\ 2 & -1 & 4-x \end{array}\right] (i) Find the real values of x for which matrix B does not have an inverse. (ii) Use the adjoint method to find the inverse of the matrix when x = -2 . (c) What are the matrices corresponding to the linear transformations in R^2 which (i) reflect a point in the x-axis, (ii) scale the co-ordinates of a point by 2, (d) Give the image, rank, kernel and nullity of the linear transformation which maps a point (r, y, z) in R^3 to the point (x, 0, z). Hence show that the Rank-Nullity theorem is satisfied.