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Exercises:.
 Let Mn(F) denote the set of all square matrices of order n, over a field of scalars, F. Prove that
Mn(F) forms a vector space over F with respect to matrix addition and scalar multiplication.
 Prove that the determinant of an upper triangular matrix is the product of its diagonal element
 Let A be a matrix where A  M n (F ) , such that det A ≠ 0. Prove that if    ( A) then 1   ( A 1 )
0  1
 Given the square matrix A  M 2 ( Z 5 ) , where A  
 , compute:
1 0 
(i ) det A
(ii ) the eigenspace for each    ( A) .
(iii ) the Jordan Canonical Form
 If A, B  M n ( F ) where A ~ B (specially, A = S-1BS, where det S  0 ), then the following
properties hold:
(i) det A = det B;
(ii) if A is diagonalizable, then so is B;
(iii) if x is an eigenvector of A, then Sx is an eigenvector of B.
 If A, B  M n ( F ) are simultaneously diagonalizable, then they have a common eigenvector.
 Prove that the eigenvalues of a real n-dimensional symmetric matrix are real, and all
eigenvectors corresponding to each eigenvalue are orthogonal.
 Solve the non-homogeneous ODE
3  1 1 
dX
 AX  F (t )
, in which A  2 0  1, F (t )  (0,0, e 2t ) T
dt


X(0)  (1,1,1) T
1  1 2 
 Apply linear transformation, transform y=4x12+6x1x2-4x22-20x1+10x2-8 into y=a0+a1t12+a2t22 ;
And find the value of a0,a1,a2.
 2 1  1
 Given A   1 4 3 , try to find eigenvalue s and eigenvecto rs, then diagonalize A


 1 3 4 