Articles of determinant

Determinant of determinant is determinant?

Looking at this question, I am thinking to consider the map $R\to M_n(R)$ where $R$ is a ring, sending $r\in R$ to $rI_n\in M_n(R).$ Then this induces a map. $$f:M_n(R)\rightarrow M_n(M_n(R))$$ Then we consider another map $g:M_n(M_n(R))\rightarrow M_{n^2}(R)$ sending, e.g. $$\begin{pmatrix} \begin{pmatrix}1&0\\0&1\end{pmatrix}&\begin{pmatrix}2&1\\3&0\end{pmatrix}\\ \begin{pmatrix} 0&0\\0&0 \end{pmatrix}&\begin{pmatrix} 2&3\\5&2\end{pmatrix} \end{pmatrix}$$ to $$\begin{pmatrix}1&0&2&1\\ 0&1&3&0\\ 0&0&2&3\\ 0&0&5&2\end{pmatrix}.$$ Is it true […]

Is there anything like upper tridiagonal matrix? How to find the determinant of such a matrix?

I want to find the determinant of the following matrix. $$\left[\begin{matrix} -\alpha_1 & \beta_2 & -\gamma_3 & 0 & 0 & 0 & \cdots & 0&0 \\ 0 & -\alpha_2 & \beta_3 & -\gamma_4 & 0 & 0 & \cdots & 0 & 0 \\0 & 0 & -\alpha_3 & \beta_4 & -\gamma_5&0&\cdots&0 & 0 […]

Blockwise Symmetric Matrix Determinant

This question arises from another one of mine, but separate enough that I feel it deserves its own thread. Wikipedia says that $$det\begin{bmatrix}A&B\\B &A \end{bmatrix} = det(A+B)det(A-B)$$ Regardless of whether or not A and B commute. Using the general formulation $$det\begin{bmatrix}A&B\\C &D \end{bmatrix} = det(A)det(D – CA^{-1}B)$$ We see that this becomes $$det(AD- ACA^{-1}B)$$ Or […]

Induction for Vandermonde Matrix

Given real numbers $x_1<x_2<\cdots<x_n$, define the Vandermonde matrix by $V=(V_{ij}) = (x^j_i)$. That is, $$V = \left(\begin{array}{cccccc} 1 & x_1 & x^2_1 & \cdots & x^{n-1}_1 & x^n_1 \\ 1 & x_2 & x^2_2 & \cdots & x^{n-1}_2 & x^n_2 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 1 & x_{n-1}& […]

Determinants and Matrices

Suppose $A$ is a $4\times4$ matrix with $\det A=2$. Find $\det((1/2) A^T A^7 I A^T A^{-1})$ where $I$ is a $4\times4$ identity matrix. My work so far: We know that $\det A^T=\det A$. $I$ has no effect on the determinant. $det A^{-1}$ is $1/\det A$. With that said, I think it looks a little like […]

Showing that $\det(M) = \det(C)$

Let $n \in \Bbb N \setminus \{0\}$ and $n_1,n_2 \in \Bbb N$ such that $n_1+n_2=n$ $$M=\begin{pmatrix}I_{n_1}&B\\O&C\end{pmatrix}$$ where $I_{n_1} \in \Bbb R^{n_1 \times n_1}$ is the identity matrix, $B \in \Bbb R^{n_1 \times n_2}$, $O \in \Bbb R^{n_2 \times n_1}$ is the zero matrix, $C\in \Bbb R^{n_2 \times n_2}$ I want to show that $\det(M)=\det(C)$ Are […]

Closed formula for $\sum_{i=1}^n \det(v_1, \ldots, Av_i, \ldots, v_n)$

This question already has an answer here: Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A =\displaystyle{ \sum_{i=1}^n} f_0(x_1, \ldots, Ax_i,\ldots, x_m)$ 5 answers

Is determinant of matrix multiplied its transpose always positive?

Assume $A$ is an arbitrary $m\times n$ real matrix. Is $\det(AA^T)$ always positive? Is it non-negative or it can have any value? Edit: It seems I have to emphasis that $m \ne n$ i.e. matrix is non-squared.

Determinant of an almost-diagonal matrix

I would like to compute the determinant of the $(k+1)\times (k+1)$ matrix below $$J=\begin{vmatrix} y_{k+1}& 0 & \ldots & 0 & y_1 \\ 0& y_{k+1}& \ldots& 0& y_2 \\ \vdots& \vdots& & \vdots &\vdots \\0 & 0&\ldots& y_{k+1} &y_k \\ -y_{k+1} & -y_{k+1} &\ldots &-y_{k+1}& \left(1-y_1-\ldots-y_k \right) \end{vmatrix} $$ The matrix is diagonal, if you […]

Derivative of the determinant of the right stretch tensor

I have to evaluate the derivative $$ \frac{\partial\det\mathcal{U}}{\partial F} $$ where $\mathcal{U}=\sqrt{F^TF}$ and $F$ is a $m\times n$ real matrix. Any suggestion would be appreciated. Thank you all, guys!! You helped me a lot.