# Michael Artin’s Algebra Ch.1 M.1

(I have started reading Michael Artin’s Algebra and I think uploading solutions of some problems, especially some of the starred ones, might be useful to others like me who are currently self-studying mathematics. If I have obtained the solution from some source, I will mention the source with the solution. Pointing out errors is always very welcome.)

Problem Statement: $M= \begin{bmatrix} A & B \\ C & D \end{bmatrix}$ where each block is an $n \times n$ matrix. Suppose that $A$ is invertible and that $AC=CA$ . Use block multiplication to prove that $det M=det(AD-CB)$.

Solution:

We note that $G=\begin{bmatrix} 0 & I_n \\ A & -C \end{bmatrix}$ is invertible, and by expanding the determinant along its first row, we see that $det G = det A$.

Then $GM= \begin{bmatrix} A & B \\ 0 & AD-CB \end{bmatrix}$.

Take $H=\begin{bmatrix} A^{-1} & 0 \\ 0 & I_n \end{bmatrix}$.

Then $HGM = \begin{bmatrix} I_n & A^{-1}B \\ 0 & AD-CB \end{bmatrix}$.

Clearly, $det M = det (HGM)$. Expanding $HGM$ along its first column, we get $det (HGM) = det (AD - CB)$.

# “A prisoner of conscience” : Dr Binayak Sen

The following is a guest post by Kisholoy Mukherjee.

Some Wall street journalist writes an article on Binayak Sen, referred to as a prisoner of conscience by Amnesty International, a doctor and civil rights activist who has been held guilty by a Raipur Court and is sentenced to life imprisonment. The charge against him – Sedition. Proof – NONE!! Nor does he verbally corroborate the violence that is done by Maoists/Naxalites. But he only says that he finds that the backdrop of such tribal and rural movements is a reality. Is that not right? Like hell it is. But just read what this article by Wall Street Journal has to say about the conviction. No where does it mention that there was no evidence of his supporting violence. Just interacting with them is a seditious act?? What has this rigid policy of not negotiating brought to this country, apart from a continued misfortune and misery? Be it Kashmir or the Naxals, what good is just trying to wish away the problem when it INDEED exists?? But in this article, that the grave situation among the indian poor is a reality is put in inverted commas…as if to suggest it is not.

Of his alleged support for the Maoists, Dr. Sen said he didn’t support violence from either the state or the Maoists but that the grievances that Maoists were tapping into for support among the populace were “real.”

The article is clearly aimed at staunch capitalists who shriek at the mention of things like humanity, civil rights, justice and equality (and of course the ‘other -ism’ {courtesy Michael Moore}). And it is also aimed at misinforming, deliberate convolution of facts and blatant lying along with propaganda of an endorsed misconception. It reads:

The insurgency that began in a village called Naxalbari in the eastern state of West Bengal in 1967 seeks to overthrow the Indian government in a bid to present a communist paradigm of development. They have attracted support by playing up local grievances such a lack of school and health facilities and the perceived abuse of land rights in the name of industrialization.

You can read it over and over again, but you still may not find anything wrong with it. Even with the highlighted parts. If that is the case, chances are, you too are one of these blood sucking, lying crony capitalists, or one who has been brainwashed to the point of idiocy and intellectual blindness by the jihad-against-socialism’s propaganda machinery. Since the highlighted parts are nothing but simply lies -a truck load of them. The tribal movements against land acquisition and similar corporate takeovers of their rights and the Naxalite movements are two different things altogether. Yes, at some point, either due to need of support or due to forceful infiltration by political parties like the Left, the tribal movement, that dates back way before 1967, and that started out as a completely social movement, out of sheer necessity, became interconnected with the Naxalite or the Maoist movement. Because of twisted media reporting, external pressure by political parties and some unscrupulous individuals, the just movement of the have-nots and the deprived assumed a political color (mostly red!). The real intention was to simply ensure that the rights of the poorest people did not get exploited and snatched away from them by the rich yet greedy. But it was never to “present a communist paradigm of development” as the article falsely claims.

Again, by using the phrase “playing up”, an attempt was perhaps made to make it appear in front the world that the problems that the movements tried to address were unreal or exaggerated. It is such a betrayal of reality to even remotely suggest something like that – the same article misses no opportunity to paint the malady of the Maoist or Naxalite movement as the “plague” of the nation, but it unfairly leaves out the other part of the equation – that the grievances of severe underdevelopment and poverty are very much a reality and they are themselves one of the biggest plagues constantly affecting the nation. And “perceived” abuse of land rights?? Yeah, right.

That the article was disproportionately biased towards one side, would be a gross understatement. It was just an example of what crony capitalism is at its worst – a manipulator of truth even to suit its personal agenda.