Linear Algebra (Part 1)

Problem 1

If A=(({{2, 1, 3}, {4, 1, 5}, {-2, 1, 3}})) and B=(({{1, -1, 1}, {3, 1, 2}, {0, 2, 1}})), find (A^T)B-(B^T)A.

Problem 2

If A=(({{20, 17, -24}, {16, 8, -16}, {28, 14, -28}})), find (A^413).

Problem 3

For what value of t will the reduced row echelon form of matrix (({{t, 1, 1}, {4, 1, -1}, {1, 1, t}}))  be equal to (Subscript[I, 3])?

Problem 4

If the determinant of (({{a, b, c}, {d, e, f}, {g, h, i}}))=17, what is the determinant of (({{2c, 2f, 2i}, {-a, -d, -g}, {b+2a, e+2d, h+2g}}))?

Problem 5

Find the determinant, transpose, trace, and inverse of (({{1, 3, 2}, {4, 1, 1}, {5, 2, 2}})) .

Problem 6

Find the inverse of the matrix (({{3, 4}, {5, 1}})) over (Subscript[ℤ, 7]).

Problem 7

Find the eigenvalues and corresponding eigenspaces for the matrix (({{1, 2}, {3, 2}})).

Problem 8

Check that λ=1 is an eigenvalue for the matrix (({{-7, -16, -8}, {6, 13, 6}, {-2, -4, -1}})) and find the corresponding eigenspace for it. (λ=3 is also an eigenvalue but you do not need to do anything with it).

Problem 9

If A=(({{0, 1, 0}, {0, 0, 2}, {1, 0, 0}})), find the sum of the eigenvalues of A and the matrix (A^24).

Problem 10

Given that λ=7+(Sqrt[55]) and λ=7-(Sqrt[55]) are eigenvalues of (({{5, 1, 6}, {7, 1, 8}, {2, 4, 9}})), what is the third eigenvalue?

Problem 11

If A=(({{0, 1, 0, 0, 0}, {1, 0, 0, 0, 0}, {0, 0, 0, 0, 1}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}})), what is (A^121)?  (note: there is a trick for this that uses a combination of linear and abstract algebra - think about what A xis, where x is a column of (Subscript[I, 5]))