Could someone help me with the proof that a square matrix has maximal rank if and only if it is invertible? Thanks to everybody

I have an inhomogenous ODE. The main issue here is variables are matrices. It is bit of matrix calculus. A solution would be highly appreciated interms of x . I guess we can use same methods for solving ODEs but have to be careful because these are matrices $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, […]

I have some weak memory that some sources I have encountered a long time ago make some connection between random walks and wavelets, but I am quite sure it is not in the same sense. What I was thinking is to in the matrices representing filtering operations replace every $a\in \mathbb{R}$: $$\cases {\phantom{-}a\to \begin{bmatrix}a&0\\0&a\end{bmatrix}\\-a\to \begin{bmatrix}0&a\\a&0\end{bmatrix}}$$ […]

I have noticed that for the case of 1×1, 2×2 and 3×3 matrices $A$, $B$, I can write the determinant of their commutator $C=[A,B]$ in terms of traces: 1×1 matrices $A$, $B$: $$\det(C)=\text{tr}(C)$$ 2×2 matrices $A$, $B$: $$\det(C)=-\frac{1}{2}\text{tr}(C^2)$$ 3×3 matrices $A$, $B$: $$\det(C)=\frac{1}{3}\text{tr}(C^3)$$ But I can’t find a simple formula for 4×4 matrices–I have no […]

So I’m studying linear algebra and one of the self-study exercises has a set of true or false questions. One of the question is this: If $A^2 = I$ (Identity Matrix) Then $A = \pm I$ ? I’m pretty sure it is true but the answer say it’s false. How can this be false (maybe […]

Let $A, B \in R^{n \times n}$. Prove that $e^{-A} = (e^{A})^{-1}$. ($R$ is the real numbers) I’ve tried messing around with both sides, evaluated as sums. I just can’t get the two to match up. Any ideas?

I already asked this question which has been answered. This question may seem very similar but the required matrix manipulations are probably very different here due to the addition of the matrix $\mathbf{P}$, which makes the problem more difficult. “Experimentally”, I found that the kernel (null space) of the following matrix is of dimension 2. […]

Let $A$ be a square matrix of order $n$ such that $A^2 = I$. Prove that if $1$ is the only eigenvalue of $A$, then $A = I$. Prove that $A$ is diagonalizable. For (1), I know that there are two eigenvalues which are $1$ and $-1$, how do I go about proving what the […]

Do anybody know how matrices are built into ZFC theory ? I pretty have no idea how to build them from ZFC axioms. I would like a constructive proof of their existence if possible. Thanks in advance.

I am working on the following problem. Let $e^{Mt} = \sum\limits_{k=0}^{\infty} \frac{M^k t^k}{k!}$ where $M$ is an $n\times n$ matrix. Now prove that $$e^{(M+N)} = e^{M}e^N$$ given that $MN=NM$, ie $M$ and $N$ commute. Now the left hand side of the desired equality is $$e^{(M+N)} = I+ (M+N) + \frac{(M+N)^2}{2!} + \frac{(M+N)^3}{3!} + \ldots $$ […]

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