Let M be a 3 3 non-zero matrix such that M^3 = O and M^2 O , where O is the null matrix. If A = I - 2M + 3M^2 and its inverse is given by A⁻¹ = I + M + M^2 for some real numbers , , , then the value of ^2 + ^2 + ^2 is equal to ______.

Step-by-step solution

Given A = I - 2M + 3M^2 and A⁻¹ = I + M + M^2 . We know that A A⁻¹ = I . (I - 2M + 3M^2)( I + M + M^2) = I Expanding the product algebraically: I + M + M^2 - 2 M - 2 M^2 - 2 M^3 + 3 M^2 + 3 M^3 + 3 M^4 = I Since it is given that M^3 = O , it follows that M^4 = O as well. Substituting these into the expansion gives: I + ( - 2 )M + ( - 2 + 3 )M^2 = I Because M^2 O and M^3 = O , the matrices I, M, M^2 are linearly independent. We can equate the corresponding coefficients on both sides: Coefficient of I : = 1 Coefficie