7
$\begingroup$

If $A$ is a $10\times10$ matrix with entries from the set $\{0, 1, 2, 3\}$ and if $AA^T$ is of the form: $$\begin{pmatrix} 0 & * & * & \cdots & * \\ * & 0 & * & \cdots & * \\ * & * & 0 & \cdots & * \\ \vdots & \vdots & \vdots & \cdots & \vdots \\ * & * & * & \cdots & 0 \end{pmatrix}$$

Then the number of such matrices $A$ is:
A) $(4^3)^{10}$ B)$(4^2)^{10}$ C)$4^{10}$ D) $1$

Since all the diagonal elements of $AA^T$ is zero I could realize that it is a skew-symmetric matrix. But, I'm no able to understand how I can use this result for finding the possibilities of the original matrix. I would like hints rather than answers.

$\endgroup$
  • 2
    $\begingroup$ Say that $A=(a_{ij})$. Then the $i$-th diagonal entry of $AA^t$ is $(AA^{t})_{ii}=\sum_{j=1}^{n}a_{ij}^2=0$. If the sum of two nonnegative numbers is zero, then the summands must be zero. Consequently, $a_{ij}^2=0$ for all $i,j$. So every entry of the matrix you started with was zero. $\endgroup$ – Adam Azzam Aug 26 '13 at 2:23
  • 1
    $\begingroup$ He asked for hints :) $\endgroup$ – Avraham Aug 26 '13 at 2:36
2
$\begingroup$

Since you mentioned symmetry, try simplifying the expression $(AA^T)^T$. What do you get, and what does this say about $AA^T$?

To find the possibilities of the original matrix, try writing the $(i,i)$-th diagonal element of $AA^T$ in terms of $A$, then equate the result with $0$.

$\endgroup$
1
$\begingroup$

I do not know if all such matrices are skew-symmetric, but if the original matrix has to be skew-symmetric, what do you know about the relationship between $a_{ij}$ and $a_{ji}$? Which members of the set {0, 1, 2, 3} exhibit that property?.

Edit

Even if the original is not skew-symmetric, try writing out $AA^T$ for a 3x3 matrix—what are you multiplying for each element of the diagonals? Try writing what the first diagonal entry would be in matrix multiplication for $AA^T$ in a 10x10. The second. What does this mean for each entry on the diagonal? Which members of the set {0, 1, 2, 3} can provide the needed main diagonal of 0's in the answer?

$\endgroup$
  • $\begingroup$ But, it is not given that the original matrix is skew-symmetric. $\endgroup$ – Rajath Radhakrishnan Aug 26 '13 at 1:54
  • $\begingroup$ You are correct. However, look at the edited portion of the above answer for a further hint. $\endgroup$ – Avraham Aug 26 '13 at 2:17
  • $\begingroup$ When I take 10*10 with general elements a,b,c.....etc and multiply it with its transpose I get the diagonal elements as the sum of the squares of the elements of the particular row. But, only zero can satisfy this condition. Thus, the original matrix is a 10*10 zero matrix. So,there is only one possibility. Is my thinking correct? $\endgroup$ – Rajath Radhakrishnan Aug 26 '13 at 12:25
  • $\begingroup$ Exactly! The diagonals are the sum of the squares of all of the elements in each row, which must be nonnegative, so every element must be 0. $\endgroup$ – Avraham Aug 26 '13 at 12:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.