# Questions tagged [transpose]

In linear algebra, the transpose of a matrix is another matrix whose i-th row and j-th column is the j-th row and i-th column of the original matrix.

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### suppose $A$ is an $m\times n$ real matrix and $A^\top A$ is $3\times3$ nonsingular matrix, then $n =$? [closed]

the answer of this question is $n<3$ , why? I mean how we decided that is $n<3$ when $A$ multiply $A^\top$ will be singular?
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### Z-matrix multiplied by its transpose

What can be said about eigenvalues and eigenvectors of $ZZ^T$ if Z is a Z-matrix whose columns sum to zero?
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### What does the transpose of a linear transformation represent? [duplicate]

I'm struggling to understand what the transpose of a linear transformation represents. My textbook's motivation for this wasn't very helpful. All it did was ask, "Is there a linear transformation ...
### Prove if $M^2 =0$, then $\operatorname{rank}(M+M^T)=2\operatorname{rank}(M)$.
Prove if $M^2 =0$, then $\operatorname{rank}(M+M^T)=2\operatorname{rank}(M)$. I have used the rank nullity theorem and the fact that rank of matrix and its transpose is same to prove it but I am not ...