MATH 110: LINEAR ALGEBRA MIDTERM #2 REVIEW

Problem 1. Mark the following statements true or false. If the statement is true, give a reason; if it is false, give a counterexample.

(a) If A, B are row-equivalent matrices, then det(A) = det(B).

(b) Let A, B be n × n matrices such that AB = O. Then rk(A) + rk(B) ≤ n.

(c) The row space of a matrix A is equal to the column space of A.

(d) If A is a matrix which is similar only to itself, then A = I.

(e) Let A be an m × n matrix with rank m. Then there exists an n × m matrix B such that BA = I, where I is the m × m identity matrix.

(f) det(A + A^{t}) = 2 det(A) for any matrix A.

(g) If a ≠ b, then the n × n-matrix

A = (\begin{matrix}a + b & ab & 0 & … & 0 & 0 \\ 1 & a + b & ab & … & 0 & 0 \\ 0 & 1 & a + b & … & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & … & a + b & ab \\ 0 & 0 & 0 & … & 1 & a + b \end{matrix})

has

det A = \frac{a^{n+1} − b^{n+1}}{a − b} .

(h) If A, B, C, D are n × n-matrices, then the 2n × 2n-matrix

M = (\begin{matrix}A & B \\ C & D \end{matrix})

has determinant

det M = (det A)(det D) − det(B) det(C).

(i) Similar matrices have the same characteristic polynomial.

(j) The matrices A and A^{t} have the same eigenvalues.

(k) If two matrices have the same characteristic polynomial and one is diagonalizable, then so is the other.

(l) A matrix A satisfies A^{2} = I if and only if the only eigenvalues of A are ±1.

(m) If A ≠ O is a matrix such that A^{k} = O for some k ≥ 1, then A is not diagonalizable.

(n) If A is an n × n-matrix and f(t) is a polynomial such that f(A) = O, then the characteristic polynomial of A divides f(t).

(o) If A is a matrix with eigenvalue λ and f(t) is a polynomial, then f(λ) is an eigenvalue of f(A).