Show that :
$$\frac{d}{dt} A^{-1}(t) = -A^{-1}(t) (\frac{d}{dt} A(t) ) A^{-1}(t) $$
A(t) is a matrix.
2026-04-02 22:13:07.1775167987
Derivative of a nonsingular matrix
204 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in LINEAR-ALGEBRA
- An underdetermined system derived for rotated coordinate system
- How to prove the following equality with matrix norm?
- Alternate basis for a subspace of $\mathcal P_3(\mathbb R)$?
- Why the derivative of $T(\gamma(s))$ is $T$ if this composition is not a linear transformation?
- Why is necessary ask $F$ to be infinite in order to obtain: $ f(v)=0$ for all $ f\in V^* \implies v=0 $
- I don't understand this $\left(\left[T\right]^B_C\right)^{-1}=\left[T^{-1}\right]^C_B$
- Summation in subsets
- $C=AB-BA$. If $CA=AC$, then $C$ is not invertible.
- Basis of span in $R^4$
- Prove if A is regular skew symmetric, I+A is regular (with obstacles)
Related Questions in MATRICES
- How to prove the following equality with matrix norm?
- I don't understand this $\left(\left[T\right]^B_C\right)^{-1}=\left[T^{-1}\right]^C_B$
- Powers of a simple matrix and Catalan numbers
- Gradient of Cost Function To Find Matrix Factorization
- Particular commutator matrix is strictly lower triangular, or at least annihilates last base vector
- Inverse of a triangular-by-block $3 \times 3$ matrix
- Form square matrix out of a non square matrix to calculate determinant
- Extending a linear action to monomials of higher degree
- Eiegenspectrum on subtracting a diagonal matrix
- For a $G$ a finite subgroup of $\mathbb{GL}_2(\mathbb{R})$ of rank $3$, show that $f^2 = \textrm{Id}$ for all $f \in G$
Related Questions in DERIVATIVES
- Derivative of $ \sqrt x + sinx $
- Second directional derivative of a scaler in polar coordinate
- A problem on mathematical analysis.
- Why the derivative of $T(\gamma(s))$ is $T$ if this composition is not a linear transformation?
- Does there exist any relationship between non-constant $N$-Exhaustible function and differentiability?
- Holding intermediate variables constant in partial derivative chain rule
- How would I simplify this fraction easily?
- Why is the derivative of a vector in polar form the cross product?
- Proving smoothness for a sequence of functions.
- Gradient and Hessian of quadratic form
Related Questions in NORMED-SPACES
- How to prove the following equality with matrix norm?
- Closure and Subsets of Normed Vector Spaces
- Exercise 1.105 of Megginson's "An Introduction to Banach Space Theory"
- derive the expectation of exponential function $e^{-\left\Vert \mathbf{x} - V\mathbf{x}+\mathbf{a}\right\Vert^2}$ or its upper bound
- Minimum of the 2-norm
- Show that $\Phi$ is a contraction with a maximum norm.
- Understanding the essential range
- Mean value theorem for functions from $\mathbb R^n \to \mathbb R^n$
- Metric on a linear space is induced by norm if and only if the metric is homogeneous and translation invariant
- Gradient of integral of vector norm
Related Questions in MATRIX-DECOMPOSITION
- Real eigenvalues of a non-symmetric matrix $A$ ?.
- Swapping row $n$ with row $m$ by using permutation matrix
- Block diagonalizing a Hermitian matrix
- $A \in M_n$ is reducible if and only if there is a permutation $i_1, ... , i_n$ of $1,... , n$
- Simplify $x^TA(AA^T+I)^{-1}A^Tx$
- Diagonalize real symmetric matrix
- How to solve for $L$ in $X = LL^T$?
- Q of the QR decomposition is an upper Hessenberg matrix
- Question involving orthogonal matrix and congruent matrices $P^{t}AP=I$
- Singular values by QR decomposition
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Differentiate with respect to $t$ both sides of the equation $$ A(t) [A(t)]^{-1} = I, $$ getting (via the product rule) $$ A(t) \frac{d}{dt} [A(t)]^{-1} + \Bigl(\frac{d}{dt} A(t)\Bigr) [A(t)]^{-1} = 0. $$ Now multiply both sides of the equation above by $[A(t)]^{-1}$ on the left, getting $$ \frac{d}{dt} [A(t)]^{-1} = - [A(t)]^{-1}\Bigl(\frac{d}{dt} A(t)\Bigr) [A(t)]^{-1}. $$