$A$ 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
- Consider the function $G(x, y, z) = 0$. This function allows us to implicitly define each of three variables as a function of the other two variables. Assume that all partial derivatives of the function $G(x, y, z)$ exist everywhere. Then, the value of $\left(\frac{\partial z}{\partial x} \times \frac{\partial x}{\partial y} \times \frac{\partial y}{\partial z}\right)$ is _________ (in integer)
- Levenshtein distance is used to measure the minimum edit distance between two strings by counting the minimum number of editing operations (such as, insertions, deletions, substitutions) required to transform one string to another. Consider an alternative version of the Levenshtein distance in which each insertion and each deletion has a cost of 1 and substitutions are not allowed.
Based on this alternative version, the Levenshtein distance between the strings, word and work is ______ (Answer in integer). - In a given collection of documents, let $N$ be the total number of documents and let $d$ be the number of documents in which the term $t$ occurs.
Which ONE of the following fractions is used to define the inverse document frequency (idf) of the term $t$? - The total number of unique character bigrams in the string thoughtsoftheghosts (which contains 19 characters) is ______.
A bacterium can be approximated as a cylinder with a hemisphere at each end, as shown in the figure. The cylinder has a height of $1 \mu\text{m}$ and diameter of $1 \mu\text{m}$. Assuming that the density of the bacterium is equal to that of water, what is the approximate mass of this bacterium?
Given: density of water = $10^3 \text{ kg/m}^3$; $1 \mu\text{m} = 10^{-6}\text{ m}$
Volume of a cylinder = $\pi r^2 h$, where $r$ is the radius and $h$ is the height of the cylinder
Volume of a sphere = $\frac{4}{3} \pi r^3$, where $r$ is the radius of the sphere
