If A= [ array rr 2 3 & - 2 3 2 3 & 2 3 array ] and I is an identity matrix of order 2 2 then A^3=

Step-by-step solution

The matrix A is given by A = bmatrix 2 3 & - 2 3 2 3 & 2 3 bmatrix . This is a rotation matrix of the form R( ) = bmatrix & - & bmatrix where = 2 3 . The property of rotation matrices states that R( )^n = R(n ) . Therefore, A^3 = R (3 2 3 ) = R(2 ) . Calculating the values for R(2 ) : R(2 ) = bmatrix (2 ) & - (2 ) (2 ) & (2 ) bmatrix = bmatrix 1 & 0 0 & 1 bmatrix . Since bmatrix 1 & 0 0 & 1 bmatrix = I , it follows that A^3 = I . Answer: I