Problem 73 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(A\) be a square matrix all of whose entries are integers. Then which one of the following is true? [2008] (a) If det \(A=\pm 1\), then \(A^{-1}\) exists but all its entries are not necessarily integers (b) If det \(A \neq \pm 1\), then \(A^{-1}\) exists and all its entries are non integers (c) If det \(A=\pm 1\), then \(A^{-1}\) exists but all its entries are integers (d) If det \(A=\pm 1\), then \(A^{-1}\) need not exists

Step-by-Step Solution

Answer

Option c is true: If det \( A = \pm 1 \), then \( A^{-1} \) exists and all its entries are integers.

Step 1: Understanding the Determinant Condition

The determinant condition, \det A = \pm 1\, is crucial in determining whether the inverse exists and the nature of its entries. A determinant of \( \pm 1 \) indicates that matrix \( A \) is unimodular, which has specific implications for its inverse.

Step 2: Determining the Existence of the Inverse

For square matrices, an inverse exists if and only if the determinant is not equal to zero. Here, since \det A = \pm 1\, the determinant is non-zero, indicating that \( A^{-1} \) does indeed exist.

Step 3: Analyzing Integer Property of the Inverse

If \( A \) is a square matrix with integer entries and determinant \( \pm 1 \), then \( A \) is invertible with \( A^{-1} \) also having integer entries due to the property of unimodular matrices. This is because unimodular matrices are both integer matrices themselves and have integer inverses.

Step 4: Conclusion Based on the Options

Given that \( A^{-1} \) exists and has integer entries when the determinant is \( \pm 1 \), option \( c \) (If det \( A = \pm 1 \), then \( A^{-1} \) exists but all its entries are integers) is correct.

Key Concepts

Integer Matrix

An integer matrix is a type of matrix where all the elements are integers. This means each entry in the matrix is a whole number, without any fractions or decimals. For example, a matrix like

• \( \begin{bmatrix} 1 & 2 \ -3 & 4 \end{bmatrix} \)

is an integer matrix because each element is an integer.
Integer matrices are significant in various mathematical contexts because they simplify calculations and have unique properties. One of these is that under certain conditions, such as having a determinant of 1 or -1, an integer matrix can have an inverse that is also an integer matrix. This feature is not shared by matrices with non-integer entries, making integer matrices particularly useful in applications where maintaining integer properties is essential.

Matrix Inverse

In linear algebra, a matrix inverse is like the reciprocal of a number. If a matrix \( A \) has an inverse, denoted as \( A^{-1} \), then:

• \( A \times A^{-1} = I \)
• \( A^{-1} \times A = I \)

where \( I \) is the identity matrix, serving the same role as 1 does in arithmetic—acting as a neutral element.
Not all matrices have inverses. For a matrix to be invertible, its determinant must not be zero. When dealing with integer matrices, if the determinant equals \( \pm 1 \), not only does the inverse exist, but it is also an integer matrix. This is because the calculation for the inverse preserves integer property due to the determinant condition.
Finding the inverse of a matrix involves using the formula:

• \( A^{-1} = \frac{1}{ ext{det}(A)} \, \text{adj}(A) \)

where \( \text{adj}(A) \) is the adjugate of \( A \). For integer matrices with \( ext{det}(A) = \pm 1 \), this division results in another integer matrix.

Unimodular Matrix

A unimodular matrix is a special type of integer matrix with a determinant of 1 or -1. These matrices have some very interesting properties. The most notable property is that their inverses are also integer matrices.
This makes unimodular matrices extremely useful in number theory and integer programming, as they help maintain integer solutions across transformations. By containing all integer entries and possessing integer inverses, unimodular matrices maintain a "closed loop" of sorts within the integer domain.
The concept of unimodularity is directly tied to the practical applications of integer matrices, particularly in algorithm design and optimization. When a matrix is unimodular, it has a stable structure which is reliable for calculations that require precision and adherence to integer values. This is why they are often used in fields where exactness is key.