Matrix Operations

  1. Let \(A=\left[\begin{array}{rr}{2} & {-3} \\ {-4} & {6}\end{array}\right], B=\left[\begin{array}{ll}{8} & {4} \\ {5} & {5}\end{array}\right], \) and \(C=\left[\begin{array}{rr}{5} & {-2} \\ {3} & {1}\end{array}\right]\). Verify that \(A B = A C \) and yet \(B \neq C\).   solution


  2. Let \(A=\left[\begin{array}{rr}{2} & {-3} \\ {-4} & {6}\end{array}\right], B=\left[\begin{array}{rr}{5} & {-2}\\ {3} & {1}\end{array}\right]\). Verify that (a) \((A + B)^T = A^T+B^T \), and (b) \((A B)^T = B^T A^T \).  solution


  3. Find two matrices A and B such that their product is a zero matrix but neither A nor B is a zero matrix.   Solution: see solution of 2.