Final answer:
To find a 2x2 matrix A such that A⁴ is the identity matrix, we can use the concept of matrix multiplication. One possible solution is a = 1, b = 0, c = 0, d = 1.
Step-by-step explanation:
To find a 2x2 matrix A such that A⁴ is the identity matrix, we can use the concept of matrix multiplication. Let's consider a general 2x2 matrix A = [a b; c d]. If we calculate A², we get [a² + bc ab + bd; ac + cd bc + d²]. If we then calculate A⁴, we get [a⁴ + 2a²bc + b²d² 2ab²c + b³d + ac³ + 2bcd² + d⁴; 2a²c + abd + c³ + 2bcd + bd³ ac² + bcd + c²d + d⁴].
For A⁴ to be the identity matrix, each element of A⁴ must equal 1 if the element is on the diagonal or 0 if the element is off the diagonal. Therefore, we need to find values for a, b, c, and d that satisfy these conditions. One possible solution is a = 1, b = 0, c = 0, d = 1. This gives us the matrix A = [1 0; 0 1], which when raised to the power of 4 is equal to the identity matrix.