Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [matrix-rank]

For questions regarding the rank of matrices in linear algebra.

In linear algebra, the rank of a matrix $A$ is the size of the largest collection of linearly independent columns of $A$ (the column rank) or the size of the largest collection of linearly independent rows of $A$ (the row rank). For every matrix, the column rank is equal to the row rank. It is a measure of the “nondegenerateness” of the system of linear equations and linear transformation encoded by $A$. There are multiple definitions of rank. The rank is one of the fundamental pieces of data associated with a matrix.

If $A$ is the matrix of a linear map $f$, then the rank of $A$ is equal to the dimension of the image of $f$.

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If $A$ and $B$ are $n^{\text{th}}$ order matrices and $\text{rank}$ of $AB$ is $1$ then $\text{rank}$ of $BA$ can't be $n$. Explain

If $A$ and $B$ are $n^{\text{th}}$ order square matrices. I think statement is true but I didn't know a clear reason and proof of it please explain. Got it as rank of AB <= mini(rank(A), rank(B)).....(1) rank(BA) <=…
Anjana Beniwal
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Rank and column/row space

The question that I am trying to solve is Suppose that A is n × n and has rank n. What is its row space? What is its nullspace? What is its column space? From rank nullity theorem, I know that the null space will be 0 because. Also, rank =…
gazillionTickets
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How to prove $rank(A^+)$ is no more than rank(A)?

Look here, my friend. How to prove the following equation? Or give a counter-example. $$\text{rank}(A^+)\leq \text{rank}(A)$$ where $A$ has full rank, and $\text{rank}(A^+)$ represents the positive components of matrix $A$,…
guorui
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When two boys have same marks, why subsequent rank is kept blank?

Trying to understand when two boys get same marks as follows, subsequent ranks is not awarded it is kept blank; is this correct interpretation in mathematical terms ? Given below, Why boy who got 88 marks not awarded 4th rank ? MARKS/RANK…
Rodney-Sydney
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Values of rank $AB$

If $A$ is a $4\times2$ matrix and $B$ is a $2\times 3$ matrix, what are the possible values of $\operatorname*{rank}(AB)$? Construct examples of $A$ and $B$ exhibiting each possible value of $\operatorname*{rank}(AB)$ and explain your reasoning.
dz_killer
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If $\operatorname{rank}(A)=\operatorname{rank}(AB)$, prove that $\operatorname{rank}(AB)=\operatorname{rank}(ABA)$

Let $A,B$ be $n\times n$ matrixes over a field $K$. If $\operatorname{rank}(A)=\operatorname{rank}(AB)$, prove that $\operatorname{rank}(AB)=\operatorname{rank}(ABA)$.
mo qian
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rank of certain block matrix need to clarified

Given that $E$ is an $n_1\times n_1$ matrix and $C$ is a $p_1\times n_1$ matrix. We are given that rank of $\begin{bmatrix}I_{n_1}-E\\C\end{bmatrix}=rank(C)$ then we need to show rank of $\begin{bmatrix}E\\C\end{bmatrix}=n_1$ Thanks for helping.
Myshkin
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Why do I get differnt row reduced echelon form?

As far as I know ,one matrix has a unique row reduced echelon form. But why do I get the different echelon form if I change the last row operation of the first picture? What have I done wrong ? What's the exact rule here so that I can get the same…
techie z
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