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.

learn more… | top users | synonyms

3
votes
6answers
180 views

Matrix inverse identity

Question: Assuming that all matrix inverses involved below exist, show that $$(\mathbf{A}-\mathbf{B})^{-1}=\mathbf{A}^{-1}+\mathbf{A}^{-1}(\mathbf{B}^{-1}-\mathbf{A}^{-1})^{-1}\mathbf{A}^{-1}$$ in ...
3
votes
1answer
106 views

Fast way to calculate determinant for a block matrix

I have a block matrix $$Q_{(n+m-1)\times(n+m-1)} = \begin{pmatrix} A & -J\\-J^t & B \end{pmatrix}$$ where $$A_{(m-1)\times(m-1)} = n*I_{(m-1)\times(m-1)} \text{ and } B_{n\times n} = ...
9
votes
6answers
32k views

What does it mean to have a determinant equal to zero?

After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$. I hope someone can explain this ...
1
vote
2answers
72 views

Matrix Identity Proof

Let $A$ and $C$ be $3 \times 2$ matrices and let $B$ be a $2 \times 2$ matrix such that $AB=C$. Prove that: $$||A_1 \times A_2 || \cdot |\det B| = ||C_1 \times C_2 ||$$ where $A_i$ and $C_i$ are the ...
1
vote
3answers
123 views

If $J$ is the $n×n$ matrix of all ones, and $A = (l−b)I +bJ$, then $\det(A) = (l − b)^{n−1}(l + (n − 1)b)$

I am stuck on how to prove this by induction. Let $J$ be the $n×n$ matrix of all ones, and let $A = (l−b)I +bJ$. Show that $$\det(A) = (l − b)^{n−1}(l + (n − 1)b).$$ I have shown that it holds ...
5
votes
3answers
333 views

Special orthogonal matrices have orthogonal square roots

Let $A$ be an orthogonal matrix with $\det (A)=1$. Show that there exists an orthogonal matrix $B$ such that $B^2=A$. Thank you very much.
0
votes
2answers
175 views

Linear algebra: need help with proof

Can someone please help me with this proof. For $A,B$ ∈ $F^{n×n}$, show that $AB$ and $BA$ have the same characteristic polynomial.
3
votes
1answer
51 views

Finding Determinants Recursively

From the MIT OCW Linear Algebra (18.06) final exam, question 9: For square matrices with 3's on the diagonal, 2s on the diagonal above, and 1s on the diagonal below: $$A_1=\begin{pmatrix} 3 ...
2
votes
2answers
155 views

Prove that if $AC^T = |A|I \implies \det C = (\det A)^{n-1}$

Prove that if $AC^T = |A|I \implies \det C = (\det A)^{n-1}$ Ran into trouble with a proof for linear algebra. $C$ is the cofactor matrix of $A \in \mathbb{R}^{n\times n}$, and I'm not sure how to ...
0
votes
1answer
50 views

Determinant formula and invertibility.

I am working on a problem where I need to find the determinant of $$ \begin{bmatrix} b & a & & \\ & b & a \\ & & & \ddots \\ & & & & ...
14
votes
2answers
167 views

Is a linear combination of minors irreducible?

Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
1
vote
1answer
36 views

Is it true that in $Mat(n,n) $ the set of singular matrices forms a hyperplane?

Is it true that in $Mat(n,n)$ the set of singular matrices forms a hyperplane, separating the matrices of positive determinant from the matrices of negative determinant? This is my intuition, but ...
3
votes
1answer
95 views

Why Vandermonde's determinant divides such determinant?

Assume that $$ W(x_1,...,x_n;k)=\left [ \begin{array}{rrrrrrrr} 1 & x_1 &... & x_1^{n-2} & x_1^k \\ 1 & x_2 &... & x_2^{n-2} & x_k \\ & & \ddots \\ 1 & ...
1
vote
0answers
70 views

Why the ith coefficient of $|\lambda I-A|$ is the sum of all $i$-th order principle minors of $A$?

I come across a theorem that $f(\lambda )=|\lambda I-A|$, which equals to $\lambda ^{n}-a_{1}\lambda ^{n-1}+\alpha _{2}\lambda ^{n-2}-...(-1)^{n}a_{n}$ where $a_{i}$ is the sums of all ith order ...
2
votes
2answers
149 views

Proof of a Determinant Identity

I have a matrix identity that I wish to prove, which relates the determinant of a matrix to determinants of sub-matrices (essentially, cofactors of the larger matrix). In general terms, consider the ...
0
votes
0answers
274 views

Determinant of a general circulant matrix

I'm dealing with a problem that is comparable to "How do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$?", yet slightly more difficult: I was asked to determine the ...
1
vote
1answer
85 views

Function defined over $\mathbb{R}^2$

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be given as $$f(x,y)= (e^x \cos y, e^x \sin y)$$ 1) Show that the determinant det D$f$ of the derivative of f is never zero. $\frac{\partial(e^x \cos ...
1
vote
4answers
59 views

Finding determinant of a simple matrix [duplicate]

Can someone please explain how to compute the determinant of $J_n - I_n$ where $j_n$ it a matrix of ones? E.g. for $n=5$ we get the following matrix $$\left(\begin{array}{ccccc} 0 & 1 & 1 ...
5
votes
4answers
310 views

Why is it true that $\mathrm{adj}(A)A = \det(A) \cdot I$?

This is a statement in linear algebra that I can't seem to understand the proof behind. For a square matrix $A$, why is: $$\mathrm{adj}(A)A = \det(A) \cdot I$$ Any explanation would be greatly ...
3
votes
1answer
231 views

Hugely ugly messy determinant - any trick to find it?

Find the determinant of $$\begin{bmatrix}1 & a & a^2 & a^{3}\\ 1 & b & b^{2} & b^{3}\\ 1 & c & c^{2} & c^{3}\\ 1 & d & d^{2} & d^{3} \end{bmatrix}$$ ...
2
votes
2answers
45 views

why this is correct: $\det(C+Di)$ is not zero, then there exists some real number $a$ such that $\det(C + a D)$ is not zero

I wonder why the following statement is correct: supposing $C$ and $D$ are two real matrix, if the determinant of the complex matrix $C + D i $ is not zero, then there exists some real number $a$ ...
10
votes
7answers
600 views

Check if $\det(I + S) = 1 + \operatorname{trace}(S)$ holds ?

I saw the following statement in my homework and we are asked to make use of the statement: If $S$ is a symmetric matrix then $$\det(I + S ) = 1 + \operatorname{trace}(S).$$ However, I am not ...
8
votes
3answers
317 views

Without choosing bases, how to show that the determinant is multiplicative in this sense?

I was recently considering this statement: Let $V$ be a finite-dimensional $k$-vector space, and let $\phi:V\to V$ be an endomorphism. Suppose that $W\subseteq V$ is a subspace that is stable ...
7
votes
3answers
247 views

Matrix Determinant Identity

I have come across an observation about the determinant of a matrix, but I don't know how to prove it in general. Let me demonstrate it through an example. $$ \begin{align} \left| \begin{matrix} 1 ...
2
votes
1answer
48 views

Calculate the determinant of the matrix

I'm asked to find the determinant of a matrix $B$ if: $$A=\left |\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ ...
1
vote
2answers
193 views

Linear Algebra True / False with justifications.

Which of the following are correct or incorrect? A) The determinant function $det:M_n\rightarrow R$ is a linear transformation. False - determinants tell us areas or volumes. B) If $A$ is $n ...
0
votes
2answers
159 views

Matrices Problem with 3 Unknown Variables J, K and M

Given: $$ \left[\matrix{1&3 \\-2&4 }\right]+ \left[\matrix{11&5 \\-6&12 }\right]=K\left(\left[\matrix{3&2 \\J&M }\right]\right) $$ Find the value of $J+K+M.$ the answer is $6$ ...
0
votes
2answers
413 views

What are actually proportional rows(columns) in a determinant?

So, I posted the question about determinant equality and thoght that two columns are proportional. But they are not. Please explain me this on the next exaple (I thought the first and the second ...
0
votes
3answers
128 views

Find a $\lambda$ so the system has a unique solution?

$$\begin{align} 3x + \lambda y & = 5 + \lambda \\ 2x + 5y & = 8 \end{align} $$ I got that $\lambda$ can be anything by using Cramer's rule, so there are infinite solutions.
3
votes
2answers
239 views

Parallelogram area using determinant

Given a Parallelogram with the co-ordinates: $(a+c, b+d), (c,d), (a, b)$ and $(0, 0)$ I have to prove that the area of the Parallelogram is: $|ad-bc|$ as in the determinant of: $$\begin{bmatrix} a ...
14
votes
2answers
384 views

easier way of calculating the determinant for this matrix

I have to calculate the determinant of this matrix: $$ \begin{pmatrix} a&b&c&d\\b&c&d&a\\c&d&a&b\\d&a&b&c \end{pmatrix} $$ Is there an easier way of ...
1
vote
1answer
89 views

Calculating determinant of a matrix whose non-zero elements are two sub-matrices on its diagonal

Given a $m \times m$ square matrix $M$: $$ M = \begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix} $$ $A$ is an $a \times a$ and $B$ is a $b \times b$ square matrix; and of course $a+b=m$. All the ...
0
votes
4answers
297 views

Relationships between $\det(A+B)$ and $A+B$

When computing $\det(A+B)$ we notice that there is no relation between $\det A + \det B$. However does the $\det(A+B)$ have any relation to the matrices $A+B$ as they stand?
4
votes
3answers
124 views

Does assigning a different inner product to a vector space in $\mathbb{R^n}$ change the meaning of the determinant on that space?

We just started talking about inner product spaces and and how one can assign a different notion of length and angle on a vector space. Since the determinant in $\mathbb{R^n}$ captures the notion of ...
11
votes
5answers
482 views

Show determinant of matrix is non-zero

I have $a,b,c\in\mathbb{Q}$ not all zero. ($a^2+b^2+c^2\ne 0$), I want to show that the following determinant is then non-zero. I failed to arrive at an appropriate form of the polynomial. Help ...
-1
votes
1answer
73 views

divisibility of polynomials and determinant relations

Let $A$ be an integral domain and $f(x), g(x) \in A[x_1,\cdots,x_n]$. Write $f(x)=\sum \alpha_{\omega} x^{\omega}, g(x) = \sum \beta_{\omega} x^{\omega}$ where $\omega = (\omega_1,\cdots,\omega_n)$ ...
4
votes
2answers
188 views

Math hack for solving system of equations

Is it a "standard" Math/Numerical-Analysis hack to add a relatively small number e.g. 1*10E-5 to the diagonal of a squared matrix to ensure LU Decomposition (or whichever decomposition algorithm is ...
2
votes
1answer
55 views

Clarifying Theorem 4.11 of Lang's Algebra textbook.

Can someone more explicitly describe Theorem 4.11 in Algebra? Let $E$ be a module over a commutative ring $R$, and let $v_1,\dots,v_n$ be elements of $E$. Let $A=(a_{ij})$ be a matrix in $R$, and ...
18
votes
4answers
868 views

A problem on Condition $\det(A+B)=\det(A)+\det(B)$

Let $A$ be a matrix $n\times n$ matrix that for any matrix $B$ we have $\det(A+B)=\det(A)+\det(B)$. If this imply that $A=0$? or $\det(A)=0$?
3
votes
1answer
108 views

eigenvalues of block matrix with the eigenvalues of one block already known

Give a matrix which can be decomposed into 4 parts $B = \left[\begin{matrix}A &I \\ -I &0\end{matrix}\right]$ where $I$ denotes the identity matrix and $0$ is a zero matrix. It's easy to ...
7
votes
4answers
2k views

How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?

Prove $$\det(e^A) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}_{n×n}$.
2
votes
2answers
159 views

Determinants and Matrices

Suppose $A$ is a $4\times4$ matrix with $\det A=2$. Find $\det((1/2) A^T A^7 I A^T A^{-1})$ where $I$ is a $4\times4$ identity matrix. My work so far: We know that $\det A^T=\det A$. $I$ has no ...
1
vote
2answers
279 views

How to prove that det($A^{T}A$) is nonnegative?

Why is the determinant of the product of a matrix and its transpose nonnegative?
2
votes
3answers
413 views

Correlation between polynomial equations and matrix determinants

Expanding $p(x)=(ax-b)(cx+d)$ we get $acx^2+(ad-bc)x-bd$. Notice the determinant of the matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix} $ is $ad-bc$ exactly like the constant of $x$ ...
2
votes
4answers
1k views

Determinant of matrix exponential?

Suppose $A$ is a $n \times n$ constant matrix. How can I prove $\det(e^A) = e^{\displaystyle \sum_{\lambda_i\in\sigma(A)} \lambda_i}$, where $\sigma(A)$ is the multiset of eigenvalues of $A$? The ...
2
votes
1answer
98 views

Question on determinants of matrices changing between integer matrices

The following problem came up from a though I had while reading: Let's say we have $M=\mathbb{Z}^n$ and we have another free $\mathbb{Z}$-module, $N$, inside of $M$ also with rank $n$. We know we ...
2
votes
2answers
200 views

Why is the Leibniz formula for determinants called such?

My professor said that Leibniz was not even aware of the concept. The Wikipedia page says that the formula was named "in honor of Gottfried Leibniz." What gives? Did he do work that was related, and ...
21
votes
4answers
829 views

Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected

Math people: In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times ...
2
votes
2answers
54 views

Determinants of related matrices.

Given: det $\begin{bmatrix} r & s & t \\ u & v & w \\ x & y & z \\ \end{bmatrix}=4$, compute det ...
1
vote
0answers
109 views

maximize an objective function with an infinite component

Suppose I have the following maximization problem: $\log\det(\alpha K_p)-c\alpha$ with respect to $\alpha$ with $c$ being a constant and $m$ being the dimension of $K_p$. Here, one of the eigenvalues ...