Articles of determinant

Eigenvalues of a certain tridiagonal matrix

Consider $A \in M_n(\mathbb R)$ defined by: $$A=\begin{bmatrix} a & -1 & 0 & \cdots & 0 \\ -1 & a & -1 &\cdots& 0\\ 0 & -1 & a & \cdots & 0\\ \vdots & \vdots & \vdots &\ddots & -1\\ 0 & 0 & 0 & -1 & a \end{bmatrix}$$ How to find […]

Possible matrix-determinant identity

Is $$\det(I+ ABB^*A^*C^{-1})=\det(I+ B^*A^*ABC^{-1})$$ where $I$ is identity matrix, $A,B,C$ are complex valued matrices. And $C$ is $(I+X)$ where $X$ is PSD. I know that this makes $ABB^*A^*$ and $B^*A^*AB$ positive semi/definite. And I know for square matrices $X,Y,Z$ we have $\det(XYZ)=\det(YXZ)$ so $\det (ABB^*A^*C^{-1})=\det(B^*A^*ABC^{-1})$. And also $\det (I + ABB^*A^*C^{-1})=\det(I + B^*A^*ABC^{-1})$. Please comment […]

Determinant form of equation, 3 variables, third order (nomogram)

I’m trying to put the following equation in determinant form: $12h^3 – 6ah^2 + ha^2 – V = 0$, where $h, a, V$ are variables (this is a volume for a pyramid frustum with $1:3$ slope, $h$ is the height and $a$ is the side of the base, $V$ is the volume). The purpose of […]

On integral of a function over a simplex

Help w/the following general calculation and references would be appreciated. Let $ABC$ be a triangle in the plane. Then for any linear function of two variables $u$. $$ \int_{\triangle}|\nabla u|^2=\gamma_{AB}(u(A)-u(B))^2+ \gamma_{AC}(u(A)-u(C))^2+\gamma_{BC}(u(B)-u(C))^2, $$ where $$ \gamma_{AB}=\frac{1}{2}\cot(\angle C), \gamma_{AC}=\frac{1}{2}\cot(\angle B), \gamma_{BC}=\frac{1}{2}\cot(\angle A). $$ What is a good reference for the formula? Is it due to R. Duffin? […]

Rank of square matrix $A$ with $a_{ij}=\lambda_j^{p_i}$, where $p_i$ is an increasing sequence

Let $$ A = \begin{bmatrix} \lambda_1^{p_1} & \lambda_2^{p_1} & \cdots & \lambda_n^{p_1} \\ \lambda_1^{p_2} & \lambda_2^{p_2} & \cdots & \lambda_n^{p_2} \\ \lambda_1^{p_3} & \lambda_2^{p_3} & \cdots & \lambda_n^{p_3} \\ \vdots & \vdots & \ddots & \vdots \\ \lambda_1^{p_n} & \lambda_2^{p_n} & \cdots & \lambda_n^{p_n} \end{bmatrix}$$ where $p_1=0$ and $p_k > p_i$ for $i < k […]

Wedge Product Formula For Sine. Analogous Formula Generalizing Cosine to Higher Dimensions?

So I was day dreaming about linear algebra today (in a class which had nothing to do with linear algebra), when I stumbled across an interesting relationship. I was thinking about how determinants are really the area spanned by column vectors, and I had the thought that one could measure linear independence (in $R^2$ in […]

Proof that the characteristic polynomial of a $2 \times 2$ matrix is $x^2 – \text{tr}(A) x + \det (A)$

Let $$ A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$$ Let $C_{A}(x) := \det(xI-A)$ be the characteristic polynomial of A. Show that $$C_{A}(x)=x^2-\text{tr}(A)x+\det(A).$$ I know that $\text{tr}(A)=a_{11}+a_{22}$ and $\det(A)=a_{11}a_{22}-a_{21}a_{12}$. Plugging this into the above equation I get $$C_{A}(x)=x^2-(a_{11}+a_{22})x+a_{11}a_{22}-a_{21}a_{12}.$$ I’m not sure how to get past this. As you can tell, I’m not too good at […]

Is this determinant identity true?

I simulated the following $$\det(I+[A|B][A|B]^*)\geq\det(I+[B][B]^*)$$ and every time I get a true result. So how can I prove this statement? Here $[A|B]$ is matrix augmentation. $I$ is the identity matrix, $A,B$ are complex valued matrices and $^*$ is the conjugate transpose operation. In my first simulation I used randomly generated complex square matrices so I […]

Calculate Determinant A size n

I am given homework like this, calculate the Matrix $$ \begin{bmatrix}x+1 &x&x&…&x\\x&x+2&x&…&x\\x&x&x+3&…&x\\…&…&…&…&…\\x&x&x&…&x+n\end{bmatrix} $$ I tried to change it into triangular matrices but the best result I get is: $$ \begin{bmatrix}x+1 &x&x&…&x\\-1&2&0&…&0\\-1&0&3&…&0\\…&…&…&…&…\\-1&0&0&…&n\end{bmatrix} $$ I wonder if I am on the right track since there seems to be no way to advanced. Any hint given is appreciate.

Finding determinant of matrix without expanding

show that determinant $$\left|\matrix{ x^2+L & xy & xz \\ xy & y^2+L & yz \\ xz & yz & z^2+L \\ }\right| = L^2(x^2+y^2+z^2+L)$$ without expanding by using the appropriate properties of determinant. All i can do is LHS $$x^2y^2z^2\left|\matrix{ 1+L/x^2 & 1 & 1 \\ 1 & 1+L/y^2 & 1 \\ 1 & […]