Problem 115 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(\mathrm{A}\) be a square matrix all of whose entries are integers. Then which one of the following is true? (A) If \(\operatorname{det} \mathrm{A}=\pm 1\), then \(\mathrm{A}^{-1}\) exists but all its entries are not necessarily integers (B) If detA \(\neq \pm 1\), then \(\mathrm{A}^{-1}\) exists and all its entries are non-integers (C) If detA \(=\pm 1\), then \(\mathrm{A}^{-1}\) exists and all its entries are integers (D) If \(\operatorname{det} \mathrm{A}=\pm 1\), then \(\mathrm{A}^{-1}\) need not exist

Step-by-Step Solution

Answer

Option C is true: If \( \det A = \pm 1 \), \( A^{-1} \) exists with integer entries.

Step 1: Understanding Determinant Conditions

A square matrix \( A \) with integer entries has a determinant \( \det A \). If \( \det A = \pm 1 \), then the matrix is invertible, meaning \( A^{-1} \) exists. If \( \det A eq \pm 1 \), the matrix may not have an inverse with integer entries.

Step 2: Exploring Option A

Option A states that if \( \det A = \pm 1 \), then \( A^{-1} \) exists but all its entries are not necessarily integers. This is incorrect because if the determinant is \( \pm 1 \), \( A \) is unimodular, hence \( A^{-1} \) has integer entries.

Step 3: Analyzing Option B

Option B suggests that when \( \det A eq \pm 1 \), \( A^{-1} \) exists and all its entries are non-integers. This is incorrect too since \( A^{-1} \) does not necessarily exist in this case.

Step 4: Evaluating Option C

Option C states if \( \det A = \pm 1 \), then \( A^{-1} \) exists and all its entries are integers. This is true as the matrix \( A \) is unimodular, and its inverse has integer entries.

Step 5: Considering Option D

Option D claims if \( \det A = \pm 1 \), \( A^{-1} \) need not exist. This is false since \( \det A = \pm 1 \) implies that \( A \) is invertible.

Key Concepts

Integer Entries

In mathematics, when we refer to a square matrix having integer entries, we mean that all the elements of the matrix are whole numbers. This is an important feature because integer entries make many calculations simpler and more manageable. For example, let's consider a 2x2 matrix:\[\begin{bmatrix}2 & 3 \5 & 7 \\end{bmatrix}\]In this matrix, all the elements—2, 3, 5, and 7—are integers. Integer matrices often arise in applications involving discrete mathematics, computer science, and other areas requiring precise, whole-count operations. When the determinant of such a matrix equals \( \pm 1 \), it is particularly noteworthy, as this special condition hints towards a unique type of matrix called a *unimodular matrix*.

Determinant

The determinant is a special scalar value associated with a square matrix. It provides important information about the matrix such as invertibility and volume scaling in transformations. For a 2x2 matrix:\[\det \begin{bmatrix}a & b \c & d \\end{bmatrix} = ad - bc\]A crucial aspect of determinants in square matrices with integer entries is that a determinant of \( \pm 1 \) guarantees that the matrix is invertible. The determinants of these matrices yield critical insights:

• If \( \det A = 0 \), the matrix is not invertible—meaning it does not have an inverse.
• If \( \det A = \pm 1 \), the matrix not only has an inverse but it has integer entries when the matrix is unimodular.

Understanding determinants is fundamental to linear algebra, and they are useful in a wide range of mathematical areas.

Unimodular Matrix

A *unimodular matrix* is a square matrix with integer entries whose determinant is \( \pm 1 \). This special type of matrix has some very compelling properties:

• It is always invertible, meaning there exists another matrix which, when multiplied with the original matrix, yields the identity matrix.
• The inverse of a unimodular matrix is likewise an integer matrix—an incredibly useful property when working in the realm of integers.
• Unimodular matrices maintain discreet and lattice-like structures, a beneficial attribute in areas like cryptography and number theory.

Given these characteristics, unimodular matrices are highly significant in mathematics, especially in solving linear equations where both coefficients and solutions are integers.

Inverse Matrix

An inverse matrix is essentially a matrix which, when multiplied with an original matrix, results in the identity matrix. For any square matrix \( A \), the inverse is denoted \( A^{-1} \). The concept of inverses is pivotal in solving matrix equations and systems of linear equations. A few key points include:

• A matrix must be square (same number of rows and columns) to have an inverse.
• A matrix has an inverse only if its determinant is not zero.
• If \( \det A = \pm 1 \), and \( A \) is unimodular, then the inverse \( A^{-1} \) will also be a matrix with integer entries.

The process of finding the inverse can be cumbersome for large matrices but remains essential for matrix algebra and applications in computational mathematics.