$A$ is an ($n \times n$) matrix. Consider the following two statements Statement 1: Columns of matrix $A$ are linearly independent Statement 2: Inverse of matrix $A$ exists Which one of the following statements is TRUE?
In this question, we are given two statements regarding an \(n \times n\) matrix \(A\) and asked to determine the logical relationship between these statements.
- We begin by analyzing **Statement 1: Columns of matrix \(A\) are linearly independent**.
- For the columns of a matrix to be linearly independent, the rank of the matrix must be \(n\), which implies that the matrix is of full rank.
- If the columns are linearly independent, it indicates that the determinant of \(A\) is non-zero.
- Next, we consider **Statement 2: Inverse of matrix \(A\) exists**.
- A matrix is invertible if and only if it is a square matrix and its determinant is non-zero.
- This condition directly implies that the matrix has full rank and its columns are linearly independent.
- To determine the relationship between the statements:
- Since both statements lead to the condition that the determinant of \(A\) must be non-zero for either to be true, they are logically equivalent.
- Thus, **Statement 1 is true if and only if Statement 2 is true**.
Therefore, the correct answer is: Statement 1 is true if and only if Statement 2 is true.
Important Questions from Numerical Computation
- An organization allows its employees to work independently on consultancy projects but charges an overhead on the consulting fee. The overhead is 20% of the consulting fee, if the fee is up to . 5,00,000. For higher fees, the overhead is . 1,00,000 plus 10% of the amount by which the fee exceeds . 5,00,000. The government charges a Goods and Services Tax of 18% on the total amount (the consulting fee plus the overhead). An employee of the organization charges this entire amount, i.e., the consulting fee, overhead, and tax, to the client. If the client cannot pay more than . 10,00,000, what is the maximum consulting fee that the employee can charge?
Three frictionless pulleys with rope attachment are in a static equilibrium as shown in the figure. The mass $m_1$ and $m_2$, in kg, respectively are
If the in-situ density of coal is 1320 kg/m$^3$ and the density of blasted coal is 952 kg/m$^3$, the swell factor is _____________ (rounded off to 3 decimal places)
A five-member truss system is shown in the figure. The maximum vertical force P in kN that can be applied so that loads on the member CD and BC do NOT exceed 50 kN and 30 kN, respectively is _____________(rounded off to 2 decimal places)
The Fourier transform and its inverse transform are respectively defined as $\tilde{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x)e^{i\omega x}dx$ and $f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \tilde{f}(\omega)e^{-i\omega x}d\omega$. Consider two functions $f$ and $g$. Another function $f * g$ is defined as
$(f * g)(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(y)g(x - y)dy$
Which of the following relation is/are true?
Note: Tilde ($\sim$) denotes the Fourier transform.
