Matrix Equation AX=0: Homogeneous Linear Equations Definition
The matrix equation $AX=0$ is a standard representation in linear algebra.
- A represents a matrix of coefficients.
- X represents a column vector of unknown variables.
- 0 represents the zero vector (a column vector with all entries equal to zero).
Understanding Homogeneity in Linear Equations
A system of linear equations is considered homogeneous if the constant term for every equation in the system is zero. When represented in matrix form $AX=B$, a system is homogeneous if and only if $B$ is the zero vector.
The given equation $AX=0$ directly fits this definition, where the vector on the right-hand side is the zero vector.
Linearity of the Equation
The equation is inherently linear because matrix multiplication ($AX$) involves only sums and products of the variables in $X$ with constant coefficients from matrix $A$. There are no terms like $x_i^2$, $x_i x_j$, or other non-linear functions.
Conclusion
Combining the properties of linearity and the zero vector on the right-hand side, the matrix equation $AX=0$ specifically represents a system of Homogeneous linear equations.
Similar Questions
- If $A \in R_{n \times n}$; $\det A=0$, then :
A. A is non singular and the rows and columns of A are linearly independent
B. A is non-singular and the rows and columns of A are linearly dependent
C. A is non-singular and A has one zero rows
D. A is singular
E. A is singular and rows and columns of A are linearly dependent
Choose the correct answer from the options given below : - The greatest rate of increase of $f = xy^{2}z^{3}$ at the point $(0, -1, -2)$ is :
- The derivative of $y = \int_{0}^{\ln x} \sin e^{t} dt$ is :
- If $\lambda = 0$, is an eigen values of A, then $\det(A)$ is :
- If A is a null Matrix, then its rank is :
- Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : If $u = xy f\left(\frac{y}{x}\right)$, then $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 2u$
Reason (R) : Given function u is homogeneous of degree 2 in $x$ and $y$.
In the light of the above statements, choose the most appropriate answer from the options given below : - Solving a second order Differential equation using Laplace transform :
A. Partial fraction expansion
B. Apply initial conditions in Laplace domain
C. Take Laplace transform of differential equation
D. Obtain time domain solution
Choose the correct answer from the options given below : - Determining Eigenvalues of a matrix :
A. Solve the characteristic equation
B. Form matrix $A - \lambda I$
C. Compute determinant
D. Identify eigenvalues
Choose the correct answer from the options given below : - A lag network for compensation normally consists of :
A. R, L and C elements
B. R and L elements
C. R and C elements
D. R only
E. L only
Choose the correct answer from the options given below : - A lead compensating network :
A. Improves response time
B. Stabilizes the system with low phase margin
C. Enables moderate increase in gain without affecting stability
D. Increases the resonant frequency
E. Decreases the resonant frequency
Choose the correct answer from the options given below :
Important Questions from Mixed Topic (CUET PG)
- Kalpsutra, the illustrated canonical text is from:-
- The Harappan city almost exclusively devoted to craft production was-:
- Mohandas Karamchand Gandhi launched quit India movement after the failure of:-
- The "Objectives Resolution" was introduced in constituent assembly by:-
- Who among the following was not a member of the constituent assembly:-
