1. Let A(t) be an anti-symmetric matrix for all t. Let M(t) be a matrix-valued function...

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• 1. Let A(t) be an anti-symmetric matrix for all t. Let M(t) be a matrix-valued function satisfying M' (t) = M(t)A(t). Show that M (t) M (t)T is a constant matrix-valued function. 2. Recall that given a matrix Ae Matn,n (R), the exponential defined by ∞o 1 k! eA = is a well-defined element of Matn,n (R). (1) Prove that (et) = AeAt. (2) Is is true that d (eB(t)) = B'(t)eB(t) dt for all matrix-valued functions B: R→ Matn,n(R)? k=0 • 3. Identify Mat2,2 (R) = R¹ and hence use the standard dot product on Mat2,2 (R). (1) Let A be an invertible matrix, A € GL(2, R) ≤ Matn,n (R). What is the angle between A and A-¹? (2) Are A and A-¹ ever perpendicular? (3) Are they ever of equal norm? and 4. Using the chain rule, compute the derivative of the maps Matn,n (R) h :Matn,n(R) A → A-3 k:Matn,n (R) A → An Hint: Write the functions as compositions of simpler functions. Matn, n (R) • 1. Let A(t) be an anti-symmetric matrix for all t. Let M(t) be a matrix-valued function satisfying M' (t) = M(t)A(t). Show that M (t) M (t)T is a constant matrix-valued function. 2. Recall that given a matrix Ae Matn,n (R), the exponential defined by ∞o 1 k! eA = is a well-defined element of Matn,n (R). (1) Prove that (et) = AeAt. (2) Is is true that d (eB(t)) = B'(t)eB(t) dt for all matrix-valued functions B: R→ Matn,n(R)? k=0 • 3. Identify Mat2,2 (R) = R¹ and hence use the standard dot product on Mat2,2 (R). (1) Let A be an invertible matrix, A € GL(2, R) ≤ Matn,n (R). What is the angle between A and A-¹? (2) Are A and A-¹ ever perpendicular? (3) Are they ever of equal norm? and 4. Using the chain rule, compute the derivative of the maps Matn,n (R) h :Matn,n(R) A → A-3 k:Matn,n (R) A → An Hint: Write the functions as compositions of simpler functions. Matn, n (R)