Question 350511
a) If A is idempotent, then {{{A^2=A}}}. You can break this down to say that {{{A*A=A}}}. From there, left multiply both sides by {{{A^(-1)}}} (you can right multiply both sides by {{{A^(-1)}}} also) to get {{{A^(-1)A*A=A^(-1)*A}}} which then becomes {{{I*A=I}}} which simplifies to {{{A=I}}}



b) The matrix A is idempotent when {{{A^2=A}}}. In order for {{{A^2=A}}} to be true, {{{A^2=A*A}}} must be defined (ie possible). So in order for {{{A*A}}} to be defined, A must have the same number of rows and columns. This means that A must be square.



c) Does {{{B^2=B}}} ? If so, then matrix B is idempotent. Notice how {{{B^2=B*B=(matrix(2,2,1,0,0,0))*(matrix(2,2,1,0,0,0))=(matrix(2,2,1,0,0,0))=B}}}. So this shows us that {{{B*B=B}}} and that {{{B^2=B}}}. So matrix B is idempotent.