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0

$begingroup$

Here is a list of very novice questions that came across while studying:

1. Suppose $A$ is an $m times n$ matrix. Is the rank of $Aleq min{m,n}$?

Attempt: The "rank" of a matrix gives us the idea of how many of the column vectors and row vectors are independent (i.e. not the linear combination of the rest of the vectors). Again, row rank$=$ column rank. So, the above relation should hold.

2. If $B$ is a non null $m times 1$ matrix and $C$ is a non null $1 times n$ matrix prove that the rank of $BC$ is $1$.

Attempt: Since both of the matrices are non null, there is at least of element in both of the two matrices, which are non zero, say $b_{i1}$ and $c_{1j}$. Hence, a minor of order $1$ exists, which is non zero. Now, the dimension of $BC$ is $m times n$. Now, the element $b_{i1}c_{1j}$ is in the matrix $BC$. Again, $1leq rank{BC}leq min{rank B, rank C}leq1$.

3. 
   $A$ is an $m times r $ matrix and $B$ is an $r times n$ matrix. If the rank of $AB$ is $m$, prove that rank of $A$ is $m$.

Attempt: $m leq min{rank A, rank B}=min{m,n}$. Again, (assuming the conclusion of $1$ is true) rank of $AB$ being $m$, it must be $mleq n$.
So, $rank A leq rank B implies rankA=m$ (otherwise $rankAB <m$)

Please do verify these three attempts.

linear-algebra matrices proof-verification matrix-rank

share|cite|improve this question

edited Mar 18 at 20:15

Subhasis Biswas

asked Mar 18 at 20:07

Subhasis BiswasSubhasis Biswas

512412

$endgroup$

add a comment | 

0

$begingroup$

Here is a list of very novice questions that came across while studying:

1. Suppose $A$ is an $m times n$ matrix. Is the rank of $Aleq min{m,n}$?

Attempt: The "rank" of a matrix gives us the idea of how many of the column vectors and row vectors are independent (i.e. not the linear combination of the rest of the vectors). Again, row rank$=$ column rank. So, the above relation should hold.

2. If $B$ is a non null $m times 1$ matrix and $C$ is a non null $1 times n$ matrix prove that the rank of $BC$ is $1$.

Attempt: Since both of the matrices are non null, there is at least of element in both of the two matrices, which are non zero, say $b_{i1}$ and $c_{1j}$. Hence, a minor of order $1$ exists, which is non zero. Now, the dimension of $BC$ is $m times n$. Now, the element $b_{i1}c_{1j}$ is in the matrix $BC$. Again, $1leq rank{BC}leq min{rank B, rank C}leq1$.

3. 
   $A$ is an $m times r $ matrix and $B$ is an $r times n$ matrix. If the rank of $AB$ is $m$, prove that rank of $A$ is $m$.

Attempt: $m leq min{rank A, rank B}=min{m,n}$. Again, (assuming the conclusion of $1$ is true) rank of $AB$ being $m$, it must be $mleq n$.
So, $rank A leq rank B implies rankA=m$ (otherwise $rankAB <m$)

Please do verify these three attempts.

linear-algebra matrices proof-verification matrix-rank

share|cite|improve this question

edited Mar 18 at 20:15

Subhasis Biswas

asked Mar 18 at 20:07

Subhasis BiswasSubhasis Biswas

512412

$endgroup$

add a comment | 

$begingroup$

Here is a list of very novice questions that came across while studying:

1. Suppose $A$ is an $m times n$ matrix. Is the rank of $Aleq min{m,n}$?

Attempt: The "rank" of a matrix gives us the idea of how many of the column vectors and row vectors are independent (i.e. not the linear combination of the rest of the vectors). Again, row rank$=$ column rank. So, the above relation should hold.

2. If $B$ is a non null $m times 1$ matrix and $C$ is a non null $1 times n$ matrix prove that the rank of $BC$ is $1$.

Attempt: Since both of the matrices are non null, there is at least of element in both of the two matrices, which are non zero, say $b_{i1}$ and $c_{1j}$. Hence, a minor of order $1$ exists, which is non zero. Now, the dimension of $BC$ is $m times n$. Now, the element $b_{i1}c_{1j}$ is in the matrix $BC$. Again, $1leq rank{BC}leq min{rank B, rank C}leq1$.

3. 
   $A$ is an $m times r $ matrix and $B$ is an $r times n$ matrix. If the rank of $AB$ is $m$, prove that rank of $A$ is $m$.

Attempt: $m leq min{rank A, rank B}=min{m,n}$. Again, (assuming the conclusion of $1$ is true) rank of $AB$ being $m$, it must be $mleq n$.
So, $rank A leq rank B implies rankA=m$ (otherwise $rankAB <m$)

Please do verify these three attempts.

linear-algebra matrices proof-verification matrix-rank

share|cite|improve this question

edited Mar 18 at 20:15

Subhasis Biswas

asked Mar 18 at 20:07

Subhasis BiswasSubhasis Biswas

512412

$endgroup$

share|cite|improve this question

edited Mar 18 at 20:15

Subhasis Biswas

asked Mar 18 at 20:07

Subhasis BiswasSubhasis Biswas

512412

share|cite|improve this question

edited Mar 18 at 20:15

Subhasis Biswas

asked Mar 18 at 20:07

Subhasis BiswasSubhasis Biswas

512412

asked Mar 18 at 20:07

Subhasis BiswasSubhasis Biswas

512412

asked Mar 18 at 20:07

Subhasis BiswasSubhasis Biswas

512412