I. (15 points) Ture or False I. Only determine true or false. No explanation required.

(a) A scalar multiple of an elementary matrix is still an elementary matrix.

(b) The union of two subspaces of a vector space V is still a subspace.

(c) If A Mmn(F ), then rank(A) m and rank(A) n.(d) If A Mnn(F ) has an eigenvalue 0 with multiplicity r, then rankA = n r.(e) The basis of the zero vector space is {0}.(f) There is no onto linear transformation from F10 to M34(F ).

II. (15 points) True or False II. Determine true or false and give explanations (i.e.,

a sketch of proof or a counterexample).

(a) If 1, 2 are distinct eignevalues of a linear operator T, then E1 E2 = {0}.(b) If A Mnn(F ) and the sum of all row vectors of A is equal to 0 Fn, then

detA = 0.

(c) If A,B Mnn(F ), then det(A+B) = detA+ detB.III. (15 points)

(a) Write down the definition of A matrix A is invertible.

(b) Suppose A,B Mnn(R) and A is invertible. Let > 0. Prove that for sufficiently large, A+B is an invertible matrix.

IV. (15 points)

(a) Write down the definition of subspaces.

(b) Prove that the subset S Mnn(F ) consisting of n n matrices A satisfyingTr(A) = 0 is a subspace of Mnn(F ).

(c) What is the dimension of this subspace? When n = 2, give a basis of this

subspace.

V. (10 points) Calculate the determinant of A =

1 2 + a 3

1 a a 13a 2 1 + a

. Herea is a complex number such that a2 = 2. Your results may contain a.

VI. (15 points) Let k be a real number. Consider the matrix

A =

3 k 1k + 1 1

M22(R). (0.1)

(a) Calculate the characteristic polynomial of A.

(b) For what values of k does A have distinct real eigenvalues? Prove your statement.

(c) For what values of k is A diagonalizable (as real matrices)?

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VII. (15 points) Let n 2. Prove that the determinant of

A =

x 0 0 0 a01 x 0 0 a10 1 x 0 a2...

......

......

0 0 0 x an10 0 0 1 an

M(n+1)(n+1)(F ). (0.2)

is equal to anxn + an1x

n1 + + a1x + a0. (Hint: Use induction on n 2. Let thedeterminant be f(n). Use cofactor expansion along the first column to find a relation

between f(n) and f(n 1).)VIII*. (Optional, 10 bonus points) Suppose A,B Mmn(F ). Prove that rank(A+

B) rank(A) + rank(B). (Hint: consider the column vectors of A + B, A, B and therelation between rank and the number of linearly independent column vectors.

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