Solve equation X times X =X for all 2x2 real-valued matrices {a_11, a_12, a_21, a_22}, where "times" means modified multiplication from the picture below, i.e., a_ij = min {a_i1+a_1j, a_i2+ a_2j}

Matrix Multiplication (26.1/25.1) Similar: C- A B, two n x n matrices k aik bki O(n3) operations replacing 99 min 99 gives c mini laik t bkil Dm) Dm-1) x W identity matrix is DO) Cannot use Strassen's because no subtraction Time is still O(n n -O(n4) Repeated squaring: 2n Wn x wn W Compute W, W2, W4,..., W2k, k log n, O(n3 log n) Matrix Multiplication (26.1/25.1) Similar: C- A B, two n x n matrices k aik bki O(n3) operations replacing 99 min 99 gives c mini laik t bkil Dm) Dm-1) x W identity matrix is DO) Cannot use Strassen's because no subtraction Time is still O(n n -O(n4) Repeated squaring: 2n Wn x wn W Compute W, W2, W4,..., W2k, k log n, O(n3 log n)