A matrix M has entries M1 = 1, M12 = 1, M21 = 2, M22 =...

 

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A matrix M has entries M₁1 = 1, M12 = −1, M21 = 2, M22 = 2. For the below, show your algebra. (a) Write down the matrix M. (b) Calculate the determinant of M. (c) Calculate the inverse M-¹ of M. (d) Calculate the determinant of M−¹. (e) For general square matrices X and Y, it holds that det(XY) = det(X) det(Y). Use that to show that det(X-¹) = (det(X))¯¹. Does that property hold for our matrix M in this problem? (f) The TRACE of a square matrix is the sum of the elements in the diagonal: tr(A) = A11 + A22 + ….. Ann for an n × n matrix A. Compute the trace of M. (g) The COMMUTATOR [A, B] of two square matrices X and Y is defined as [X,Y] = XY - YX. (1) When [X, Y] = 0 we say that X and Y commute. When [X, Y] ‡ 0, then X and Y do not commute. In general matrices do not commute. but when they do it is has important physical consequences, for example in Quantum Mechanics. Compute [M, MT]. A matrix M has entries M₁1 = 1, M12 = −1, M21 = 2, M22 = 2. For the below, show your algebra. (a) Write down the matrix M. (b) Calculate the determinant of M. (c) Calculate the inverse M-¹ of M. (d) Calculate the determinant of M−¹. (e) For general square matrices X and Y, it holds that det(XY) = det(X) det(Y). Use that to show that det(X-¹) = (det(X))¯¹. Does that property hold for our matrix M in this problem? (f) The TRACE of a square matrix is the sum of the elements in the diagonal: tr(A) = A11 + A22 + ….. Ann for an n × n matrix A. Compute the trace of M. (g) The COMMUTATOR [A, B] of two square matrices X and Y is defined as [X,Y] = XY - YX. (1) When [X, Y] = 0 we say that X and Y commute. When [X, Y] ‡ 0, then X and Y do not commute. In general matrices do not commute. but when they do it is has important physical consequences, for example in Quantum Mechanics. Compute [M, MT].