Sir, I Filled Out the CAT 2020 Online Application, However I Selected Only IIM ABC.. now I Am Worrie

No, You wo not get calls from the IIMs you have not selected based on your current CAT application. But you can refill the form with a different email ID and mobile number along with repaying the CAT application fee. Your other profile details being the same as in earlier application wo not be an issue.

1. Does $ABC = ADC implies B = D$? [closed]

Let $Ain k^ntimes m$ and $Cin k^dtimes s$ and consider the linear map $$phi^C_A: k^mtimes dto k^ntimes s phi^C_A(X)=AXC$$Then, $dimker phi^C_A=operatornamerkCcdotdimker A (d-operatornamerkC)n$.This is best seen if we look at the matrices as linear functions $f_AinoperatornameLin(k^m,k^n)$, $f_CinoperatornameLin(k^s,k^d)$, and likewise $phi^C_A(X)=f_Acirc Xcirc f_C$ becomes a function from $operatornameLin(k^d,k^m)$ to $operatornameLin(k^s,k^n)$. Consider a subspace $Wsubseteq k^s$ such that $Woplusoperatornameim f_C=k^s$. Then, the map $$Psi:kerphi^C_Ato operatornameLin(operatornameim f_C,ker f_A)oplus operatornameLin(W,k^n)Psi(X)=(left.X

ight

vert_operatornameim f_C, left.X

ight

vert_W)$$ is linear and bijective (use the fact that $X$ satisfies $gcirc Xcirc f=0$ if and only if $operatornameimleft. X

ight

vert_operatornameim fsubseteqker g$).So, for any matrix $B$, the matrices $X$ such that $AXC=ABC$ form an affine subspace in $k^mtimes d$ of positive dimension as soon as either $A$ is singular or $C$ is not full-row-rank

2. Why is Slacker Cats on ABC Family?

I thought it was weird too. Actually, ABC is owned by Disney....so it really shocked me that they would have that sort of show on. I do not mind it, but you should have seen the look on my 7 year old sister's face when she saw it. I think maybe they are just trying to get a wider range of viewers.

3. Can you recall any of the three TV networks (ABC, CBS, AND NBC) ever saying anything positive about Donald Trump or his administration?

I would say that all the networks listed are overly cautious in their reporting of trump's tantrums and lack of ethics. Words are chosen carefully, to avoid fomenting disdain for trump. Given whom they have to report on, the caution and restraint are positives.

4. In triangle abc ad and bc are altitudes prove that ar (dec) /ar(abc) =dc2/ac2?

Let AD & BE be the altitudes to sides BC & AC respectively./_ADB=/_BEA both being =90 and thus Quad ABDE is cyclic. Therefore, /_EDC=/_A & /_DEC=/_B. This makes triangles ABC and DEC similar whence we claim that Area (DEC)/Area(ABC) = DC/AC. QED.

5. HELP!!! Calling all fans of abc's LOST!?

I looked it up there this season's new ones (more significant the background survivors) Guy's name is Cesar and the women with Sayid is called Ileana

6. for all the open minded tolerent big tent liberals what do you think about abc airing a program for socialist?

If it's all such a socialist plot, why was the time put up for sale to the highest bidder in the first place? Sounds to me like ABC was behaving like a bunch of good capitalists

7. What are your ABC favourite names?

Boys and Girls Albie / Amelie Blake / Brielle Cade / Caitlin Dylan / Demi-Rose Elias / Esmae Finley / Freya Grayson / Gabriella Harlan / Harper Imari / Isla Josiah / Jasmine Kairus / Kyra Lathan / Lilly Mason / Maisie Nathan / Nyla Orion / Olivia Phoenix / Poppy Reo / Rhea Samuel / Saffron Theon / Thea Wade / Willow Zackai / Zianne For Girls I found it hard choosing one name for A and S as i like so many Ariana, Amara, Amber, Anais, Ave (AH-vay), Abigail ,Sapphire, Sienna, Skye, Scarlett, Soraya the list continues and it was between two E names Esmae and Elise. For Boys i found it hard chossing one nsme for Z i like to many Zaiden, Zachary, Zeke, Zade, also it was between two E names Elijah and Elias My top 5 out of your names Boys: Maxon (is different) Zachary Elijah Theo Callum Girls: Isla Elissa (very pretty) Olivia Harmony Rose X

8. How show that $ABC$ is equilateral?

This problem gave me a really hard time, lots of steps are involved.Step 1. If $ABC$ is equilateral and $MAF,MBD,MCE$ have the same area, then $M$ is the center of $ABC$.Given that $[x,y,z]$ are the trilinear coordinates of $M$, the areas of our triangles are proportional to: $$fracxyxz,fracyzxy,fracxzyz$$ hence $M=,1,$ is the only chance.Step 2. If the areas of $MAF,MBD,MCE$ are equal, then $M$ is the centroid of $ABC$.Take an affine map $Phi$ that brings our original triangle into an equilateral one. Since the affine maps preserve the ratio between the areas, $Phi(M)$ is the centroid of $Phi(ABC)$ due to Step1, hence $M$ is the centroid of $ABC$ and $D,E,F$ are the midpoints of the sides.Call now $m_a,m_b,m_c$ the lengths of the medians through $A,B,C$ and $a,b,c$ the lengths of the sides $BC,AC,AB$ respectively.Step 3. If $a>b$, then $m_ab$, then $fraca2fracm_a3>fracb2fracm_b3$.This is the crucial part. It is straightforward to check that the inequality is equivalent to: $$ a^2b^2-ab > 2c^2-4m_a m_b,$$ but since $a^2b^2geq 2ab$, it is sufficient to prove the weaker inequality: $$2c^2-ableft(fracc2fracm_c3

ight)frac2m_a3,$$ hence the triangles $MAF,MBD$ and $CEF$ cannot all have the same perimeter, contradiction.