Articles of transpose

Prove $A^tB^t = (BA)^t$

Just going over some old homework problems for my test tomorrow. One of the questions was the prove $A^tB^t = (BA)^t$ and at the time I was really unsure of my answer and wrote the following: My Answer: We can write the $ij^{th}$ entry of $(BA)$ as $(BA)_{ij} = \sum_{k=1}^m{b_{ik}a_{kj}}$. Coincidentally, $(A^tB^t)_{ij}=\sum_{k=1}^n(A^t)_{ik}(B^t)_{kj} = \sum_{k=1}^na_{ki}b_{jk}$ Is […]

To create a special matrix !!

How to create a $N \times N$ matrix with $1$ and $-1$ as its elements, such that when this matrix is multiplied with its transpose the resultant matrice is $N \times \mathbb{I}_N$. Where $N$ is a scalar and $\mathbb{I}_N$ is the $N \times N$ identity matrix. Examle for 2×2 matrix $-1$ $1$ $1$ $1$ Example […]

$A^TA$ is always a symmetric matrix?

Through experience I’ve seen that the following statement holds true: “$A^TA$ is always a symmetric matrix?”, where $A$ is any matrix. However can this statement be proven/falsefied?

Transpose of block matrix

I’m attempting to prove that $$ \left[ \begin{array}{c c} A & B \\ C & D \\ \end{array} \right]^\top = \left[ \begin{array}{c c} A^\top & C^\top \\ B^\top & D^\top \\ \end{array} \right]. $$ Intuitively, I can see that it’s true. However, when I try to formally prove it, I quickly get lost in the […]

What is the geometric interpretation of the transpose?

I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the transpose, or even symmetric matrices. For example, if I have a linear transformation, say on the plane, my intuition is […]

Is the rank of a matrix the same of its transpose? If yes, how can I prove it?

I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view: “The rank of a matrix A is the number of non-zero rows in the reduced row-echelon form of A”. The lecturer then explained that if the matrix […]

A rank-one matrix is the product of two vectors

Let $A$ be an $n\times m$ matrix. Prove that $\operatorname{rank} (A) = 1$ if and only if there exist column vectors $v \in \mathbb{R}^n$ and $w \in \mathbb{R}^m$ such that $A=vw^t$. Progress: I’m going back and forth between using the definitions of rank: $\operatorname{rank} (A) = \dim(\operatorname{col}(A)) = \dim(\operatorname{row}(A))$ or using the rank theorem that […]

Transpose of inverse vs inverse of transpose

I can’t seem to find the answer to this using Google. Is the transpose of the inverse of a square matrix the same as the inverse of the transpose of that same matrix?

Assistance with proof of $(AB)^T=B^T A^T$

If $A$ is $m \times n$ and $B$ is $n \times p$ matrices, prove that $(AB)^T = B^T A^T$. Matrices’ elements are $A = [a_{ij}], B = [b_{ij}]$. Let $C=AB=[c_{ij}]$, where $c_{ij} = \sum_{k=1}^n a_{ik}b_{kj}$, the standard multiplication definition. We want $(AB)^T = C^T = [c_{ji}]$. That is the element in position $j,i$ is $\sum_{k=1}^n […]

Help with proving that the transpose of the product of any number of matrices is equal to the product of their transposes in reverse

Specifically I am trying to show that (An)T = (AT)n where A is an mxm square matrix and n is a positive integer. This is where I’m stuck: To prove the theorem I would like to show that ((An)T)ij = ((AT)n)ij for all ij. All I can think of is expanding the definition of matrix […]