Artin Problems: Section 1.M

Problems from Chapter 1: Matrices, Miscellaneous Problems

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Problem 1.M.1 :

Let a $2n \times 2n$ matrix be given in the form

$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$

where each block is an $n \times n$ matrix. Suppose $A$ is invertible and $AC = CA$ . Use block multiplication to prove that

$det M = det (AD-CB)$ . and give an example to show that this doesn’t hold if $AC \neq CA$

Problem 1.M.2 :

Let $A$ be an $m \times n$ matrix with $m < n$ . Prove that $A$ has no left inverse by comparing to $A$ to the square $n\times n$ matrix obtained by adding $m-n$ rows of zeros to the bottom.

Problem 1.M.3 :

The trace of a square matrix is the sum of it’s diagonal entries. Show the following properties of the trace.

(a) $trace(A+B) = trace(A)+trace(B)$

(b) $trace(AB) = trace(BA)$

(c) If $B$ is invertible then $trace(A) = trace(BAB^{-1})$

$trace(A+B) = \sum\limits_{i=1}^n a_{ii}+b_{ii} = \sum\limits_{i=1}^n a_{ii}+\sum\limits_{i=1}^n b_{ii}$ $= trace(A)+trace(B)$ thus (a)

$trace(AB) =$ $\sum\limits_{i=1}^n ( \sum\limits_{j=1}^n a_{ij}\cdot b_{ji})=$ $\sum\limits_{i=1}^n ( \sum\limits_{j=1}^n b_{ij}\cdot a_{ji})= trace(BA)$ therefore (b)