Problem 23 - A Complete Resource Book in Mathematics for JEE Main

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding Determinant and Inverse

A determinant, denoted as det \(A\), can help determine if a matrix is invertible. A matrix is invertible if and only if its determinant is non-zero.

Step 2: Evaluating When Inverse is Integer

If det \(A = \pm 1\), then the matrix \(A\) has an inverse that is also a matrix with integer entries. This is a property of unimodular matrices, which are defined to have integer coefficients and a determinant of \(\pm 1\).

Step 3: Checking Each Option

- Option A states that \((\det A = \pm 1) \Rightarrow A^{-1}\) exists but its entries are not necessarily integers. This is incorrect, as shown in Step 2.- Option B states that if \(\det A eq \pm 1\), \(A^{-1}\) exists and its entries are non-integers. This is incorrect because if \(\det A eq \pm 1\), \(A^{-1}\) may not even exist if \(\det A = 0\).- Option C states that \((\det A = \pm 1) \Rightarrow A^{-1}\) exists and all its entries are integers. This is correct based on the properties of unimodular matrices.- Option D incorrectly claims that \(A^{-1}\) need not exist even if \(\det A = \pm 1\), which contradicts the definition of invertibility.

Step 4: Conclusion

Based on the steps above, the only correct choice is (C). If det \(A = \pm 1\), then \(A^{-1}\) exists and all its entries are integers.

Key Concepts

Determinants

Determinants are a key concept when working with matrices. They are scalar values that are computed from a square matrix, and they reveal important properties of the matrix. Determinants help you determine if a matrix is invertible, which means if an inverse matrix exists.
For a square matrix \( A \), the determinant is often denoted as \( \det(A) \).

• If \( \det(A) eq 0 \), then the matrix is invertible. This means you can find another matrix, called \( A^{-1} \), that when multiplied by \( A \) gives the identity matrix.
• If \( \det(A) = 0 \), then the matrix is not invertible. No inverse matrix exists in this case.

Usually, determinants can be calculated using various methods such as cofactor expansion or row reduction. They offer important insights not only for invertibility but also in understanding the properties and behaviors of matrices, especially those used in linear transformations.

Matrix Inverse

Matrix inverse is a concept applied to square matrices and involves finding another matrix that, when multiplied by the original matrix, results in the identity matrix. This other matrix is known as the inverse matrix and is denoted by \( A^{-1} \).
The process of finding the inverse is crucial in many areas such as solving systems of linear equations.

• For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the inverse can be found using the formula \( \frac{1}{\det(A)} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \), provided the determinant \( \det(A) eq 0 \).
• The concept can be extended to larger matrices, though the calculations become more complex and often require numerical methods or algorithms.
• Depending on the value of the determinant, the inverse matrix will either have integer entries, non-integer entries, or may not exist at all.

Understanding the matrix inverse is vital for applying concepts in real-world scenarios such as in engineering, physics, and computer science.

Unimodular Matrices

Unimodular matrices are a special class of integer matrices with determinant \( \pm 1 \). These matrices hold significant importance because they possess unique and beneficial properties.
A unimodular matrix has integer entries, and more pleasingly, its inverse also consists entirely of integers.

• One defining property of unimodular matrices is that if \( \det(A) = \pm 1 \), \( A \) is invertible, and its inverse \( A^{-1} \) is also an integer matrix.
• This characteristic makes them ideal for applications where integer solutions are required, often encountered in linear Diophantine equations or lattice problems.
• Another advantage is that they preserve volume under linear transformations, a property useful in geometry and other areas involving transformations.

Unimodular matrices simplify computations in integer settings and are especially relevant when working with mathematical problems where maintaining integer solutions is essential.