# Tagged Questions

Question about determinants, computation or theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.

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### Is it mathematically correct to replace 1 with I?

For example, $3=3I$ $=3\begin{bmatrix} 1 & 0 \\ 0 &1 \\ \end{bmatrix}$ $=\begin{bmatrix} 3 & 0 \\ 0 &3 \\ \end{bmatrix}$ ...
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### Find a matrix with determinant equals to $\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$

Assume I have 4 matrices $A,B,C,D\in\Bbb{R}^{n\times n}$. I want to build a matrix $E\in\Bbb{R}^{m\times m}$ such that: $$\det{(E)}=\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$$ under the following ...
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### Determine the eigenvalue of a real matrix

I think to this question for two days : Let $A$ be a $3\times3$ real matrix such that $\det(A) = 1$ and $A^{-1}= A^T$. Prove that one of the eigenvalues is equal to $1$. I used the fact that ...
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### Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)

Throughout, $A$ and $B$ denote $n \times n$ matrices over $\mathbb{C}$. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = \...
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### A determinant associated to point sets in the plane

Consider $n$ distinct points in the plane $z_1, \ldots, z_n$. Form the matrix $D$ containing their squared distances as entries: $$D_{ij} \ = \ |z_i - z_j|^2 \, .$$ Obviously, this matrix is ...
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### Maximum and minimum of determinant of matrices with entries from $\{0,1\}$ or $\{-1,0,1\}$

Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the set $\bf{\{0,1\}}$. Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the ...
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### evaluation of determinant without expanding

If $\;\det \begin{pmatrix} a & x & x & x \\ x & b & x & x \\ x & x & c & x \\ x & x & x & d \end{pmatrix} =f(x)-xf’(x)$ where $f'(x)$ denotes ...
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### Expressing the determinant of a sum of two matrices?

Can $$\det(A + B)$$ be expressed in terms of $$\det(A), \det(B), n$$ where $A,B$ are $n\times n$ matrices? # I made the edit to allow $n$ to be factored in.
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### What is the determinant of exp(matrix)? [duplicate]

Given a square matrix $A$, form the Lie series of it, which is defined by: $$e^A = I + A + \frac{1}{2} A^2 + \frac{1}{3!} A^3 + \cdots + \frac{1}{n!} A^n = \sum_{k=0}^\infty \frac{1}{k!} A^k$$ Is ...
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### Is there a concept of “Cross determinant”?

Suppose $A = \begin{bmatrix}a & b \\ c & d \\\end{bmatrix}$. The determinant of $A$ is $$\det A = ad - bc.$$ Suppose $B = \begin{bmatrix}e & f \\ g & h \\\end{bmatrix}$. Now one could ...
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### Calculating determinant of matrix $n\times n$ [closed]

Given $$M := \mbox{diag} (1, 2, \dots, n) - n \, I_n + n \,1_n 1_n^T$$compute the determinant of $M$.
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### Given integers $m,n$, find integers $a,b,c$ such that $a^3+b^3+c^3-3abc=m n$

With a³+b³+c³-3abc=m (m-random integer) And a³+b³+c³-3abc=n(n-another integer) How to find a³+b³+c³-3abc=mn(m and n are Co prime) I came across this in an online math contest. Hint was given as ...
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I have a matrix of order $3 \times 3$. When I take its determinant it give 1700 from all the rows and columns except row 2. I don`t know whats going on \begin{bmatrix}1&20&0\\0&0&10\... 1answer 22 views ### Find conditions on x and y which guarantee that one can locally solve the following for u(x, y) and v(x, y) My understanding of this question is that I need to show that the following equations can be solved where u and v can be written as a function of x and y. xu^2+yv^2=9 xv^2-yu^2=7 I ... 0answers 14 views ### Sum of multiplication of non intersecting elements in a matrix Is there any formula for sum of multiplication of non-intersecting elements in a matrix? For example for a 3\times 3 matrix \begin{align*} \begin{bmatrix} a_{11} &a_{12} &a_{13}\\ a_{21} &... 3answers 70 views ### Prove by induction that \det(A^T) = \det (A) [closed] If A is an n\times n matrix then \det(A^T) = det(A) . Prove by induction that the matrix obtained by deleting the i^{\rm th} row and j^{\rm th} column of A^T is the transpose of the ... 1answer 30 views ### Determinant of complex matrix from real and complex parts Is is possible to find the absolute value of the determinant of a complex matrix M, given two real matrices A & B of the same dimension of M; where for N = 2, M would be: \begin{matrix} a_{00}... 7answers 3k views ### If A is a 2x2 matrix, what is det(4A) in terms of det(A) If A is a 2\times2 matrix, what is \det(4A) in terms of \det(A)? This seems trivial, but I'm not sure exactly what they are asking. I'm guessing I have some matrix A = \begin{bmatrix}a&... 3answers 36 views ### Using row operations to compute the following 3x3 determinant Use row operations to compute the following determinant \begin{bmatrix}3&3&-3\\3&4&-4\\2&-3&-5\end{bmatrix} I know how to easily compute the determinant using i - j + k ... 3answers 67 views ### Computing the 4 \times 4 determinant of a matrix Compute the determinant of\begin{bmatrix}1&-2&5&2\\0&0&3&0\\2&-4&-3&5\\2&0&3&5\end{bmatrix}$$by first expanding along the first row (at ... 0answers 69 views ### Determinant – graphical derivation of formula? I know how to compute determinants and I'm familiar with the geometrical meaning of determinant as the scaling factor of a unit (point/square/cube/hypercube)'s area/volume by applying a linear ... 1answer 27 views ### Is there a simple way of calculating determiant of “reverse arrowhead” matrix? I have some problems with finding determinant of the following matix. It seems easy but I keep getting something wrong. I have tried simple Laplace expansion on the first row, but I have a feeling I ... 2answers 60 views ### Prove an equality of complex matrixes If A \in M_2(\mathbb{C}) a matrix so that$$\det\left(A^2 + A + I_2\right)=\det\left(A^2 - A + I_2\right)=3 \tag1$$then$$A^2\left(A^2 + I_2\right)=2I_2. \tag2$$I tried to use Cayley-Hamilton ... 0answers 25 views ### Proof Determinant of Block Matrix does not depend of a variable I have the following matrix (called BRM):  BRM = \begin{bmatrix} -A & A & \mathbb{0}_{3\times3} & B & \mathbb{0}_{3\times1} & \mathbb{0}_{3\times1} \\ -C &... 2answers 3k views ### Coefficients of characteristic polynomial of a matrix For a given matrix A, and J\subseteq\{1,...,n\} let us denote by A[J] its principal minor formed by the columns and rows with indices from J. If the characteristic polynomial of A is x^n+... 0answers 46 views ### Given det(A) = 2. Find the Determinant of this Matrix I've run into a roadblock that my textbook doesn't seem to be able to help me with. I am not understanding how to solve these type of questions. I am assuming to receive the answer given, you do some ... 2answers 257 views ### Determinant of a 2\times 2 block matrix I would like to know the proof for: The determinant of the block matrix\begin{pmatrix} A & B\\ C& D\end{pmatrix} equals (D-1) \det(A) + \det(A-BC) = (D+1) \det(A) - \det(A+BC), when A ... 0answers 35 views ### Prove that the determinant of a triangular matrix only has one non zero permutation term Using permutations explain how for a triangular matrix only one term can be non zero. Please do not include any proofs using the cofactor method. Edit (OP's attempt as written in the comment ... 4answers 44 views ### Find the determinant using colum or row operations I find problem in simplification. When I tried to simplify I ended up doing the regular process of finding the determinant value. The matrix is \begin{pmatrix} 1 & 1 & 1 \\ a & b & c \... 1answer 42 views ### To distinguish among the various subsets of M_n(\Bbb R) I am having problem in doing a certain type of problems relating to matrices: To distinguish among the various subsets of M_n(\Bbb R) such as symmetric, diagonal, diagonalizable, upper triangular, ... 1answer 618 views ### Idiotic determinant mistake? I need to calculate$$\begin{vmatrix} \lambda & -1 & 0 & 0\\ -1 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & -1 \\ 0 & 0 & -1 & \lambda \end{vmatrix}.$$For ... 2answers 58 views ### Finding the determinant of the 5x5 matrix but can't put it in lower triangular form How to find the determinant of this 5x5 matrix? I can't put it in Lower or Upper Triangular form so I'm confused. I dont really know how to use laplace expansion \begin{bmatrix}3&0&0&3&... 3answers 155 views ### How much can we tell about \det(X) if we know \det(I + X)? What can we tell about \det(X) if we know \det(I + X)? Will it give some kind of bound for \det(X)? In general, if we know the determinant of matrix A + X, where A is a constant matrix, how ... 3answers 52 views ### How to find the determinant of the matrix? if \det \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} = 1 then \det \begin{bmatrix}a - 6g&7b - 42h&c - 6i\\d&7e&f\\g&7h&i\end{bmatrix} = ? ... 1answer 31 views ### Calculating the eigenvalues [closed] I'm trying to understand the dynamics of the eigenvectors and the eigenvalues. My question is about formula for finding the eigenvalues. At 4:15(the athor starts the calculating at 1:30) of the given ... 4answers 120 views ### Matrix equation A^2+A=I when \det(A) = 1 I have to solve the following problem: find the matrix A \in M_{n \times n}(\mathbb{R}) such that:$$A^2+A=I$$and \det(A)=1. How many of these matrices can be found when n is given? Thanks in ... 1answer 32 views ### Using determinants to find a recursive sequence I am trying to compute a three diagonal determinant in order to find the recursive relation. Let \Delta_{n}=\begin{vmatrix} 11 & 3 & 0 & 0 & \dots & 0\\ 13 & 11 & 3 &... 1answer 60 views ### How to find the determinant of a 5 x 5 matrix? Finding a 3x3 matrix is easy, but how can I find the determinant of this 5x5 matrix?? I just need an example of the first couple steps to mimic A = \begin{bmatrix} 7&1&9&-4&3\\... 1answer 54 views ### Prove that determinant of a 2x2 symmetric positive definite matrix is positive by “completing the square” method. From my understanding, determinant = product of Eigen values. Since it is a positive definite matrix, the eigen values are positive and hence, the determinant is ... 2answers 84 views ### \det{\begin{bmatrix} A & B \\ C & D\end{bmatrix}}=(D-1)\det(A) +\det(A-BC) ? [duplicate] I found this identity in Wikipedia, When D is a 1 \times 1 matrix, B is a column vector, and C is a row vector, then$$\det{\begin{bmatrix} A & B \\ C & D\end{bmatrix}}=(D-1)...
I want to prove matrix $C$ is invertible: $$C=I-A^TB(B^TB)^{-1}B^TA(A^TA)^{-1},$$ where $I$ is an identity matrix of appropriate dimensions, and $(A^TA)^{-1}$ and $(B^TB)^{-1}$ imply both $A$ and $B$ ...
### How to show the matrix $\left( \binom{x-i}{j-1}\right)_{1\leq i,j\leq 2r+1}$ has determinant (-1)^r and it's inverse?
After playing around in mathematica, I found that the matrix $\left( \binom{x-i}{j-1}\right)_{1\leq i,j\leq 2r+1}$ has determinant $(-1)^r$ for the first few $r$'s. How can I prove this this, or at ...