-1
votes
0answers
18 views
1
vote
1answer
27 views

Finding that values k that make this matrix invertible without using the determinant

The matrix in question is A = [(1,1,1),(1,2,k),(1,4,k^2)]. I know that I can row reduce the matrix to rref, which should in theory leave me with some k values in the matrix from which I can see what ...
0
votes
1answer
114 views

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 ...
1
vote
1answer
38 views

Proving that there is no invertible matrix with zero row sums using determinants

I have the following question which I know I should use the determinant to solve. Here it is: Determine if there exists an invertible $3\times3$ matrix $A$ such that $$\begin{align*} ...
0
votes
4answers
43 views

Set of all matrices with determinant 0, non-zero

I was assigned this problem in class: Let $f: M(n, \mathbb R) \rightarrow \mathbb R $ be given by $f(X) = det(X)$. Identify the sets $f^{-1}(0)$ and $f^{-1}(\mathbb R^*)$, where $\mathbb R^*$ denotes ...
0
votes
2answers
111 views

Inverse of a sum of positive definite matrices

Let $A,B$ be symmetric positive definite matrices. Let $A^{-1} = LL^T$ (Cholesky decomposition, $L$ is lower-triangular). I think the following identities are true, but I haven't found them online: $$ ...
1
vote
2answers
108 views

Explicit formula for inverse of upper triangular matrix inverse

I have $n \times n$ upper triangular matrix $A$ such as $$ \begin{bmatrix} x_1 & x_2 & \ldots & x_n \\ 0 & x_1 & \ldots & x_{n-1} \\ \vdots & \vdots & ...
0
votes
2answers
65 views

Linear Algebra Review Questions

So I have a test on Monday and my professor posted a couple of non-graded review questions that she said we should look over. Anyhow, I have a couple of questions that I'd like answered if that's ...
0
votes
0answers
36 views

Determinant after rank 1 update of a singular matrix

The rank-1 update to the inverse of a matrix and rank-1 update to the determinant of a matrix are closely related. I would like to compute the determinant of a rank-1 updated singular (rank-1 ...
2
votes
2answers
320 views

Intuition/Understanding of Inverse and Determinants

This is not homework, but extends from a proof in my book. EDIT We're given an $m \times m$ nonsingular matrix $B$. According to the definition of an inverse, we can calculate each element of a ...
6
votes
3answers
269 views

Which is easier to work out: determinant or inverse?

Suppose $A\in M_n(R)$ be a $n\times n$ matrix over some ring $R$. Which of the following two tasks is easier? to work out $\det(A)$; to work out $A^{-1}$. More specifically, I want to know the ...
3
votes
2answers
53 views

Linear Algebra determinant and rank relation

True or False? If the determinant of a $4 \times 4$ matrix $A$ is $4$ then its rank must be $4$. Is it false or true? My guess is true, because the matrix $A$ is invertible. But there is ...
1
vote
1answer
100 views

Invertibility of matrix with each element equal to cofactor

I am doing an exercise book which has one problem that asks you to prove the nonsingularity of a matrix if each element of the matrix equals its cofactor (the determinant submatrix by deleting the ...
1
vote
3answers
91 views

Determinant of a $4\times4$ invertible matrix

Let $A$ be a $4$ by $4$ invertible matrix, such that $\det(3A)=3\det(A^4)$. Then $\det(A)=3$. Would somebody please give me some clues on this? Thanks
3
votes
2answers
284 views

Finding inverse of a $3\times 4$ or $4\times 3$ matrix

Now I have no problem getting an inverse of a square matrix where you just calculate the matrix of minors, then apply matrix of co-factors and then transpose that and what you get you multiply by the ...