# 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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### Finding a Matrix from Determinants

I've stumbled upon this problem on my homework, and I have no clue how to do it, and haven't found any help online: If I'm understanding this correctly, then $det(M) = ad - cb + eh - gf$ ? What I ...
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### matrices vector spaces

Consider the vector space of 3 by 3 matrices with real coefficients. Let W denote the subset of matrices with determinant 0. Decide whether W is a subspace or not.
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### Why adding a row with another row in square matrix A doesn't change the $\det(A)$ value?

Why adding a row with another multiplied row in square matrix $A$ doesn't change the $\det(A)$ value?
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### Need help with a determinant problem [closed]

I'm learning determinants and just came across a problem. I've been trying really hard to solve it but no success so far. I just know that the answer is (3) 1 but don't know how to solve it? Please ...
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### Matrices over $\Bbb{R}$ with $2\times 2$ skew-symmetric blocks

For any complex number $z\in{\Bbb C}$, define a $2\times 2$ matrix $\hat z$ as $$\hat z:=\begin{pmatrix} a&-b\\ b&a \end{pmatrix}$$ where $z=a+ib$, $a,b\in{\Bbb R}$. Let $A=(z_{ij})$ be an ...
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### $LDL^T$ decompositon of a symmetric matrix and a matrix determinant expression for the lower triangular entries

Let $n$ be a positive integer, and let $M$ be an integral, symmetric, nonsingular matrix. As $M$ is nonsingular, there exists an $LDL^T$ decomposition such that $D = (d_j)$ is diagonal and ...
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### Do we have $\det=e^1\wedge\cdots\wedge e^n$?

If we think of the determinant as a multilinear map from the set of $n$-column vectors to $\mathbb{R}$, $$\det:\mathbb{R}^n\times\cdots\times\mathbb{R}^n\to\mathbb{R},$$ am I right in saying that ...
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### Series Expansion of the determinant for a matrix near the identity.

The problem is to find the second order term in the series expansion of the expression $\mathrm{det}( I + \epsilon A)$ as a power series in $\epsilon$ for a diagonalizable matrix $A$. Formally we ...
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### Is the function that maps a matrix to the determinant of a submatrix continuous?

Let $M$ be the space of $m \times n$ matrices over $\mathbb{R}$. For each $A$ in $M$ let $A'$ be a fixed submatrix of $A$. Is the function $M \to \mathbb{R}$ defined by $A \mapsto \det(A')$ ...
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### Recursive equations with matrices, and a question about determinants in relation to power of matrices

If we have the matrix equation $AX^{(i)} = X^{(i+1)}$ where $A$ is a constant matrix, this is what we'd call a recursive function; in matrix form. Moreover, if $X^{(i+1)} = X^{(i)}$, i.e. $AX = X$ ...
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### Show that a 2x2 matrix A is symmetric positive definite if and only if A is symmetric, trace(A) > 0 and det(A) > 0

I need to show two parts of the implication are true. First: if $A$ is $2\times 2$ and is symmetric positive definite then $trace(A)>0$ and $\det(A)>0$. Second: if $trace(A)>0$ and ...
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### Show that any orthogonal matrix has determinant 1 or -1 [duplicate]

Hello fellow users of this forum: Show that for any orthogonal matrix Q, either det(Q)=1 or -1. Thanks
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### Determinant of a matrix with 2x2 blocks

I have a matrix, say $A$ and want to find it's determinant $detA$. A is $L\times L$ and made up of $2\times 2$ blocks $M_{i,j}$ giving it a total size of $2L \times 2L$. The entries of the blocks ...
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### Square root of determinant equals determinant of square root?

Is it true that for a real-valued positive definite matrix $X$, $\sqrt{\det(X)} = \det(X^{1/2})$? I know that this is indeed true for the $2 \times 2$ case but I haven't been able to find the answer ...
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### Prove that Det(A-E)=0 if and only if AC=C

We have some $n \times n$ matrix $A$ and $n \times 1$ vector C. Let $E$ be the identity matrix. $$Det(A-E)=0 \iff AC=C.$$ Me and a few friends have been trying to prove it, but none of us could. ...
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If $A$ is any matrix and $B$ is a rank $2$ matrix of the same dimension then it follows that for any real $t$, $det(A -B) = [1-\partial_p + \frac{1}{2}\partial_p^2 ]det(A + pB) \vert _{p=0}$ I ...
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### If $A,B$ are square matrices and $A^2=A,B^2=B,AB=BA$, then calculate $\det (A-B)$

If $A,B$ are square matrices and $A^2=A,B^2=B,AB=BA$, then calculate $\det (A-B)$. My solution: consider $(A-B)^3=A^3-3A^2B+3AB^2-B^3=A^3-B^3=A-B$, then $\det(A-B)=0\vee 1\vee -1$ The result of ...
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### Compute a determinant [closed]

I want to compute this determinant: $$\begin{vmatrix} \sin(2x)&\sin(3x)&\sin(4x)\\ \sin(3x)&\sin(4x)&\sin(5x)\\ \sin(4x)&\sin(5x)&\sin(6x) \end{vmatrix}$$
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### Are determinants functions, numbers or matrices?

Let $M$ be a matrix such that $$M = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$$ As I understand it, \det(M) = \begin{vmatrix} a & b \\ ...