# Tagged Questions

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### Eigenvalues and Determinants of Two Matricies

Suppose $B=[v,e]$ is an $n \times 2$ matrix with $v=[v_1,...,v_n]^T$ and $e=[1,...,1]^T$, and $J_{2\times 2}=[(0,1),(1,0)]$, and so $Rank(BJB^T)=2$. How can we prove that $BJB^T$ and $JB^TB$ have the ...
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### Determinant of the sum of an identity matrix and a rank-two-symmetric matrix

Suppose $I$ is an $n \times n$ identity matrix, and $S$ is the $n \times n$ symmetric matrix with rank equals two. I was reading something saying that: $$\det(I-S)=(1-\lambda_1)(1-\lambda_2)$$ where ...
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### Shortest and most elementary proof that the product of an $n$-column and an $n$-row has determinant $0$

Let $\bf u$ be any column vector and $\bf v$ be any row vector, each with $n \geq 2$ arbitrary entries from a field. Then it is well known that ${\bf u} {\bf v}$ is an $n \times n$ matrix such ...
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### Is finding a matrix out of some set with a given determinant a hard problem?

Given $n\ge 2\ \ ,\ u,v,k\$ integers. Decision problem : Does a $n\times n$ - matrix with entries from $u$ to $v$ with determinant $k$ exist? In which complexity class is this problem ? Is it ...
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### Is det(A) maximal, if det(A+E) is maximal?

Let A be a binary matrix of size n x n and E be the matrix of the same size with all entries $1$. Proof or disproof : If det(A+E) has the maximal possible value, then det(A) also has the maximal ...
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### Verification Matrices & Linear Equations Part 2

...Continued Question 3 A - True because if it equals 4 then there will be infinite solutions B - True because any gradient except for one that is equal (4) will intersect giving a unique ...
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### Matrix Equation- solution

Sir, We have given $A= \begin{bmatrix}q_1 & q_2&q_3 \\ q_4 & q_5&q_6\\ q_7 & q_8&q_9 \end{bmatrix} \tag 1$. A is a matrix with determinant 1,orthogonal , invertible and ...
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### Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
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### Simple proof that a $3\times 3$-matrix with entries $s$ or $s+1$ cannot have determinant $\pm 1$, if $s>1$.

Let $s>1$ and $A$ be a $3\times 3$ matrix with entries $s$ or $s+1$. Then $\det(A)\ne \pm 1$. The determinant has the form $as+b$ with integers $a$,$b$ and it has to be proven that $a>0$ if ...
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### Determinant of a matrix shifted by m

Let $A$ be an $n\times n$ matrix and $Z$ be the $n\times n$ matrix, whose entries are all $m$. Let $S$ be the sum of all the adjoints of $A$. Then my conjecture is $\det(A+Z)=\det(A)+Sm$ , in ...
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### Properties of Determinant of matrix sum/multiplication

!Hey there :) I am currently working on a topic in control engineering and I'm currently looking for some way to relate determinants of matrix combinations to the determinant of the elements. ...
Consider an $n \times n$ matrix $A$ with the property that the row sums all equal the same number $s$. Show that $s$ is an eigenvalue of $A$. [Hint: Find an eigenvector] My attempt: By definition: ...