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 :
Matrix Determinant Condition Analysis
We are given a square matrix $A$ of size $n \times n$ from the set of real numbers ($A \in R_{n \times n}$). The key information provided is that the determinant of this matrix is zero.
The condition is stated as: $ \det A = 0 $
Singular Matrix Definition
In linear algebra, the determinant of a matrix provides critical information about its properties. Specifically:
- A square matrix is defined as singular if its determinant is exactly zero.
- Conversely, a square matrix is non-singular (or invertible) if its determinant is non-zero.
Deriving Matrix Properties
Given that $\det A = 0$, the matrix $A$ must be singular according to the definition.
Evaluating Multiple Choice Options
Let's analyze the options based on the fact that $A$ is singular:
- Option A states "A is non singular...". This contradicts $\det A = 0$.
- Option B states "A is non-singular...". This also contradicts $\det A = 0$.
- Option C states "A is non-singular...". This contradicts $\det A = 0$.
- Option D states "A is singular". This aligns perfectly with the condition $\det A = 0$.
- Option E states "A is singular and rows and columns of A are linearly dependent". While this statement is factually true for a singular matrix, Option D is the most direct and fundamental conclusion derived solely from $\det A = 0$.
Final Conclusion
The condition $\det A = 0$ directly and solely implies that the matrix $A$ is singular.
Similar Questions
- The matrix equation $AX=0$, represents :
- 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:-
