# Tagged Questions

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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### For what values of k is this singular matrix diagonalizable?

So the matrix is the following: \begin{bmatrix} 1 &1 &k \\ 1&1 &k \\ 1&1 &k \end{bmatrix} I've found the eigan values which are $0$ with an algebraic multiplicity of $2$ ...
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### Matrix diagonalisable in R, but not in C.

I know is quite easy to find a matrix $A\in\mathbb{R}^{2,2}$ that is diagonalisable if the base field is $\mathbb{C}$, but not diagonalisable if the base field is $\mathbb{R}$. The easiest example can ...
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### Largest Singular Value / Singular Value

I was wondering, what if the eigenvalues of a matrix A are all negative. So does that simply mean there is no singular value for this particular matrix?, hence I can't calculate the conditional number ...
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### Notation for replacing a matrix column with a vector

Let $A$ be an $n\times n$ matrix. Let $v$ be an $n\times 1$ matrix. Is there a notation to signify replacing the $j$-th column of $A$ with $v$? If not, what is the accepted way to denote this?
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### When should matrices have units of measurement?

As a mathematician I think of matrices as $\mathbb{F}^{m\times n}$, where $\mathbb{F}$ is a field and usually $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{C}$. Units are not necessary. However, ...
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### Finding eigenvalues/vectors of a matrix and proving it is not diagonalisable.

I have got the following matrix. $$\begin{pmatrix} -7 &4 \\ -9 &5 \end{pmatrix}$$ I need to find the eigenvalues, eigenvectors and $\textbf{prove}$ that it is not diagonalisable. I have ...
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### In how many ways can a $5 \times 5$ matrix be formed such that sum of row elements and column elements are $4$ and entries are $0$ or $1$?

Let we have a $5 \times 5$ matrix and the elements can be either $0$ or $1$ and the sum of elements of each row and column is $4$ then in how many ways can the matrix be formed ? I tried doing it in ...
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### Proving that $\operatorname{rank}(AB)$ is smaller or equal to $\operatorname{rank}(B)$ [duplicate]

I am struggling with proving the theorem that if $A$ and $B$ are $n\times n$ matrices, then: $$\operatorname{rank}(AB)\leq \operatorname{rank}(B)$$ Could anyone suggest me a hint? Any help is ...
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### Using detA and detB to calculate the determinant of matrix C

If we have C=($A^t$)$^2$BA$^3$B$^-$$^1A^-$$^3$ and detA=-2 and detB doesnt equal 0, how do we calculate det C? I know that the transpose of a matrix does not affect the determinant. Does this mean ...
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