# Tagged Questions

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### The effect of exchanging two rows on the determinant of a matrix

Answer: is -6 because of the following:
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### Determinants of Matrices det(4A) equals?

Suppose A is a 4 x 4 matrix such that det(A) = 1/64. What will det(4A^-1)^T be equal to? Here's my thinking, det(A^T) = det(A) I has no effect on the determinant. And det(A^-1) = 1/det(A) so ...
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### Matrix with entries from $1$ to $16$, each occuring once, and determinant $40800$

In OEIS, it is claimed, that the largest possible determinant of a $4\ x \ 4$-matrix with the entries from $1$ to $16$, each occuring once, is $40800$. Unfortunately, the article does not mention a ...
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### Determinant-like expression for non-square matrices

I'm interested in whether for any real matrix of size $m \times n$ there is a real number with the following properties: It is a polynomial expression with real coefficients in the entries of the ...
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### Is there an easy way to find the sign of the determinant of an orthogonal matrix?

I just learned that if a matrix is orthogonal, its determinant can only be valued 1 or -1. Now, if I were presented with a large matrix where it would take a lot of effort to calculate its ...
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### determinant of a standard magic square

What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order n ? Are there singular standard-magic-square-matrices of any order ...
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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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### Least number not being the determinant of a set of matrices

Let n > 1 be a natural number and u < v integers. How can I determine the least natural number not being the determinant of some n x n - matrix with integers in the range u..v without calculating ...
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### Gauss Seidel Method - How do I avoid calculating $L^{-1}$?

I'm trying to write a matlab code that gets a diagonal dominant matrix $A$, vector $b$, and finds an approximate solution $x$ to $Ax=b$ using Gauss-Seidel Method. I understand the theory. Suppose ...
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### Minimum absolute determinant of a regular latin square matrix

It is easy to show that a latin square of size n x n has a determinant, which is a multiple of $\large \frac{n^2(n+1)}{2}$, if n is odd and $\large \frac{n^2(n+1)}{4}$, if n is even. This is a lower ...
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### Probability that a random integer matrix is singular

Let A be a nxn-matrix with integers in the range $u..v$ , where $u<v$ are arbitary integers. Is there a formula, or at least, a good estimate, for the probability that the matrix is singular ? ...
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### Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
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### Determinant of complex matrix

How is the determinant of a complex matrix calculated? Is it the same algorithm as for real matrices, but the determinant itself is complex instead of real? (I was unable to find any hints with ...
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