Tutor profile: Bishwajit C.

Consider an $$n \times n$$ matrix $$A = (a_{i,j} )$$ with $$a_{1,2} = 1, a_{i,j} = 0 \forall (i, j) \ne (1, 2)$$. Prove that there is no invertible matrix $$P$$ such that $$P AP^{ −1}$$ is a diagonal matrix.

we have characteristic equation of $$A$$ is $$det(xI-A) = x^n$$. Since the roots of the characteristic polynomial are multiple roots hence $$A$$ is not diagonalizable. Hence A is not similar to any diagonal matrix.

Let $$a_n = 1\dots1$$ with $$3^n$$ digits. Prove that $$a_n$$ is divisible by $$3a_{n-1}$$.

We have $$a_n = 10^{3^{2n-2}}a_{n-1} + 10^{3^{n-1}}a_{n-1}+ a_{n-1}.$$ Now $$10\equiv 1(mod 3)$$. Hence $$10^{3^{2n-2}}+10^{3^{n-1}}+1\equiv 0(mod 3)$$. i.e. $$3 $$ divides $$10^{3^{2n-2}}+10^{3^{n-1}}+1$$ and hence $$3a_{n-1}$$ divides $$a_n$$.

Is the Ideal $$ I= (X+Y,X-Y) $$ a prime ideal in $$\mathbb{C}[X,Y]$$ ?

Notice that in $$\mathbb{C}[X,Y]/ I$$, we have $$X+Y=X-Y=0$$. Hence$$X=Y$$. Again, $$(X+Y)+(X-Y)=0$$. i.e. $$2X=0.$$ Hence $$X=Y=0$$ Hence we have $$\mathbb{C}[X,Y]/ I$$ is isomorphic to $$\mathbb{C}$$ which is a field and hence an integral domain. Hence $$I$$ is a prime.