# Tagged Questions

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### Block matrix determinant

I have encountered an statement several times while proving determinant of a block matrix. $$\det\pmatrix{A&0\\0&D}\; = \det(A)det(D)$$ where $A$ is $k\times k$ and $D$ is $n\times n$ ...
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### Does there exist $B$ for which $BB^T=I$?

My question is Does there exist a real matrix $B_{n\times m}$ with $m<n$ for which $BB^T=I_n$? Why do I need this? Suppose we are given a real matrix $Q_{m\times n}$ (again, with ...
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### Find signature of symmetric block matrix, given the diagonal blocks are positive / negative definite - Check my proof

This may be a basic question, but I'd like someone to double check it. We are given the matrix $A=\begin{pmatrix} A_1 & C \\ C^T & A_2\end{pmatrix}$ where $A_1$ is a $k$ by $k$ positive ...
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### Determinant of $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$

Calculate the determinant of the following matrix: $M \in M_{2n}(\mathbb{C})$ such that $$M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$$ I find that that $\det M = 2^n$ is that ...
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### Determinant of block matrix with certain properties

So I have the following 2N $\times$ 2N block matrix $H=\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ where each block in an N$\times$N matrix. Each block have the following ...
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### deteminant of a block skew-symmetric matrix

If I have a matrix if the form \begin{pmatrix} A & -B \\ B & A \end{pmatrix} how do i turn it into something like \begin{pmatrix} X & Y \\ 0 & Z \end{pmatrix} so the determinant is ...
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### How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$

Let $\mathbf{A}$ and $\mathbf{B}$ be $k \times k$ matrices and $\mathbf{M}$ is the block matrix $$\mathbf{M} = \begin{pmatrix}0 & \mathbf{B} \\ \mathbf{A} & 0\end{pmatrix}.$$ How to prove that ...
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I have a block matrix $$Q_{(n+m-1)\times(n+m-1)} = \begin{pmatrix} A & -J\\-J^t & B \end{pmatrix}$$ where $$A_{(m-1)\times(m-1)} = n*I_{(m-1)\times(m-1)} \text{ and } B_{n\times n} = ... 2answers 176 views ### Linear algebra: need help with proof Can someone please help me with this proof. For A,B ∈ F^{n×n}, show that AB and BA have the same characteristic polynomial. 4answers 85 views ### Determinant of a 4\times4 matrice with one unknown? I have to calculate the determinant of this matrice. I want to use the rule of sarrus, but this does only work with a 3\times3 matrice:$$ A= \begin{bmatrix} 1 & -2 & -6 & u \\ ...
Let $A,B \in \mathbb{R}^{n,n}$. Now $C = \begin{pmatrix} A & iB \\ -iB & A \end{pmatrix}$ and $D = \begin{pmatrix} A & B \\ -B & A \end{pmatrix}$. Show that $\det(C) \in \mathbb{R}$ ...
Trying to build a reduction from the maximum coverage problem to my research problem, I'm facing this difficulty : Let $X$ be a $n \times m$ binary matrix (with $m > n$), can we define a square ...