Solve [(x + 3)D2 − (2x + 7)D + 2]y = (x + 3)2ex
Solve xr + 2p = (9x + 6)e3x+2y , where r = ∂2z/∂x2 and p = ∂z/∂x
Evaluate the following integral
Mark each of the following statements True (T) or False (F). (Need NOT to give reasons.)
(1) A real square matrix may have complex eigenvalues and complex eigenvectors.
(2) Let M be a symmetric matrix. If M is invertible, then M−1 is also a
symmetric matrix.
(3) LetM be a real square matrix of size n . If || Mx ||2 =|| x ||2 for all x∈n , then
M is an orthogonal matrix, MTM = In .
(4) Let M be an m×n matrix, m ≠ n . We have rank(MTM ) = rank(MMT ) .
(5) Let M be an m×n matrix, m ≠ n . We have nullity(MTM ) = nullity(MM T ) .
Let Im and In be identity matrices of sizes m and n , respectively, where we
assume m > n . Can you find an m×n matrix A and an n×m matrix B such
that AB = Im and BA = In ? (Explain your answer)
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