Data Structure

Question

Assume that there are two lower triangular matrices P and Q of size $m\times m$.If matrix P and transpose of Q are fit into rectangular matrix R of size $m\times\left ( m+1 \right )$

then

(A) R[i,j+1]=Q[i,j]

(B) R[j,i+1]=Q[i,j]

(C) R[j+1,i]=Q[i,j]

(D)None of above

Answer 1

First we do transpose of Q here . 

So Q[j,i]  =   Q[i,j] 

Now after doing transpose operation , Q has become upper triangular matrix . 

Now the thing is contents of lower triangular matrix P will remain as it is in the R , so the indices of P will remain as it is in R as well.. 

As a result we have to store the elements of Q one column right in R w.r.t to original column in upper triangular matrix Q .

Hence

R[j,i+1]   =   Q[j,i]   where Q[j,i] represents element of upper triangular matrix Q after doing transpose operation on it.Thus w.r.t. original matrix Q which is lower triangular matrix : 

R[j,i+1]   =   Q[i,j] 

Hence B) should be the correct option .

Comments

If it will be square matrix

Answer will be Q[i,j]=R[j,i]

but it is rectangular matrix of m*m+1

So, if we take a matrix

1  0  0  0

2  3  0  0

4  5  6  0

 

After transpose it will be

1  2  4

0  3  5

0  0  6

0  0  0

then how answer matching with B)?

I am not getting actually

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P and Q are two different matrix. Both are lower triangular.

Now we need to merge them together and get R.

P:- a

     b  c

     d  e   f

Q:- 1

      2   3

     3    4   5

Merge to get R:- a 1 2 3

                        b c 3 4

                        d  e f  5

In R ,you see Q transpose has been taken,now we know in transpose row and colm number will be swapped so Q[i][j] should become R[j][i] but you can see that in the R along with trasnpose of Q,the colmn of Q in  shifted by 1.SO add 1 to column

Q[i][j] = R[j][i+1]

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I made this in hurry.Hope you get the point @ srestha 

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No

see for 4 in ur Q matrix

It is R[i][j] comes from Q[j][i] simply

how R[j][i+1] ??

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4 in Q is Q[3][2],i=3,j=2

4 in R is [2.4] so ,i=2=initial j of Q, j=4 = initial i of Q + 1

Answer 2

So, answer is option (B)