Adjoint of an adjoint of a matrix

Can you please help me on this question? $\DeclareMathOperator{\adj}{adj}$ $A$ is a real $n \times n$ matrix; show that: $\adj(\adj(A)) = (\det A)^{n-2}A$ I don’t know which of the expressions below might help $$ \adj(A)A = \det(A)I\\ (\adj(A))_{ij} = (-1)^{i+j}\det(A(i|j)) $$ Editor’s note: adjoint here refers to the classical adjoint. Answer I would discourage you … Read more

Is there a proof that a matrix is invertible iff its determinant is non-zero which doesn’t presuppose the formula for the determinant?

Proofs of the fact that a matrix is invertible iff its determinant is non-zero generally begin by saying “Define the determinant to be [very complicated formula]. We will now prove the result…”. This is obviously unsatisfactory to many people. Other proofs begin by listing axioms that the determinant should verify, and then prove that such … Read more

Determinant of a 3×3 matrix with trig.

So I’m not sure if this is a simple question to solve, but I was going through some exam review for my upcoming Linear Algebra exam and I came across this question. What is the determinant of the following matrix? |111sin2(a)sin2(b)sin2(c)cos2(a)cos2(b)cos2(c)| We never went over anything like this in lecture, so I am at a … Read more

What is the determinant of a complex of vector spaces?

I first met the notion of determinants of complexes of vector spaces in the book “Discriminants, Resultants, and Multidimensional Determinants“, but I just cannot understand the definition in that book. Could anyone explain it clearly or give some good references? Answer The appendix A of the book that you mention is probably the best reference … Read more

Determinant of $I-2itA$

Suppose A is an $n \times n$ positive defnite definite matrix. I want to show that $$\det(I_{n}-2itA)=\prod_{i=1}^{n} (1-2it\lambda_{i})$$ where $\lambda_{i}$ are the eigenvalues of A. I can’t seem to be able to show this! Thanks for your help. Answer For the case where $A$ is diagonalisable, we can consider the eigendecomposition of matrix $A$, as … Read more

Why does a determinant always give the same value for expanding about any row or column? [closed]

Closed. This question does not meet Mathematics Stack Exchange guidelines. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for Mathematics Stack Exchange. Closed 2 years ago. Improve this question The expansion of determinant by different row and column always gives same value. why does it true? … Read more

sum of a non singular symmetric matrix and the matrix of its eigenvalues is invertible

I am looking for a proof for this lemma: Assume Φ is a real, symmetric and non-singular matrix of order T×T with non-negative elements; Let define B=(u1,…,uT) as a column matrix of the eigenvectors of Φ and ΓT=diag(γ1,…,γT) is a diagonal matrix that consists of the corresponding eigenvalues of Φ. Then Φ+ΓT is an invertible … Read more

Solving nnth order determinant

I have a determinant of nth order that I am not able to convert into a triangular shape. I believe that this determinant is quite easy, but I can’t find a way to fully convert one of the corners into zeros. My other idea was to use the Laplace principle, but that didn’t work as … Read more