List questions from matrix-rank

Published on 15 May 2023 - 17:30

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rank decomposition

Published on 16 May 2023 - 2:01

61

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Determinant with full rank or rank deficient matrix

Published on 19 May 2023 - 8:14

137

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Prove: if $\text{rank}(A)=\text{rank}(A^2)$ then $Ax=0$ and $A^2x=0$ have the same set of solutions.

Published on 22 May 2023 - 4:44

98

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When is ${\rm rank}(A+B)={\rm rank}(A)+{\rm rank}(B)$ true?

Published on 22 May 2023 - 7:58

131

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Proving that $A=A(B−I)$ and $B=B(A−I)$ implies $A$ is diagonalizable.

Published on 22 May 2023 - 13:01

70

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question about systems of linear equations

Published on 22 May 2023 - 19:25

26

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Low rank function decomposition — necessary and sufficient conditions

Published on 27 Mar 2026 - 8:40

147

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Is the set of matrices with rank at most $2$ closed?

Published on 23 May 2023 - 5:36

66

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Rank of partitioned matrix

Published on 26 Mar 2026 - 22:13

68

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Proof of $\operatorname{rank} \left( X’ V X \right) = \operatorname{rank} (X)$

Published on 30 Mar 2026 - 3:56

124

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$W$ is a vector space of all real $n\times n$ matrices $X$ such that $AX=0$ where $A$ is a real $n\times n$ matrix of rank $r$. Find dimension of $W$.

Published on 26 May 2023 - 13:46

69

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Rank and similarity transformation of an idempotent matrix

Published on 25 Mar 2026 - 11:23

59

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An $n\times n$ matrix, $n\ge 2$ with characteristic polynomial $x^{n-2}(x^2-1)$

Published on 25 Mar 2026 - 14:34

114

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Let $A_{m\times n},B_{n\times m}$ be matrices, where $n<2m$ and $\text{rank}(A)=\text{rank}(B)=m$. Prove: $ABA\neq O$

Published on 04 Jun 2023 - 6:32

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Another way to find kernel and image using submatrix

rank decomposition

Determinant with full rank or rank deficient matrix

Prove: if $\text{rank}(A)=\text{rank}(A^2)$ then $Ax=0$ and $A^2x=0$ have the same set of solutions.

When is ${\rm rank}(A+B)={\rm rank}(A)+{\rm rank}(B)$ true?

Proving that $A=A(B−I)$ and $B=B(A−I)$ implies $A$ is diagonalizable.

question about systems of linear equations

Low rank function decomposition — necessary and sufficient conditions

Is the set of matrices with rank at most $2$ closed?

Rank of partitioned matrix

Proof of $\operatorname{rank} \left( X’ V X \right) = \operatorname{rank} (X)$

$W$ is a vector space of all real $n\times n$ matrices $X$ such that $AX=0$ where $A$ is a real $n\times n$ matrix of rank $r$. Find dimension of $W$.

Rank and similarity transformation of an idempotent matrix

An $n\times n$ matrix, $n\ge 2$ with characteristic polynomial $x^{n-2}(x^2-1)$

Let $A_{m\times n},B_{n\times m}$ be matrices, where $n<2m$ and $\text{rank}(A)=\text{rank}(B)=m$. Prove: $ABA\neq O$

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