Problem 13 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\mathrm{B}\) is a \(3 \times 3\) matrix such that \(\mathrm{B}^{2}=0\), then det. \(\left[(\mathrm{I}+\mathrm{B})^{50}-50 \mathrm{~B}\right]\) is equal to: \(\quad\) Online April \(\left.\mathbf{9}, \mathbf{2 0 1 4}\right]\) (a) 1 (b) 2 (c) 3 (d) 50

Step-by-Step Solution

Answer

The determinant is 1.

Step 1: Understanding the Problem

We have a matrix \(\mathbf{B}\) such that \(\mathbf{B}^2 = 0\). \(\mathbf{I}\) is the identity matrix. We are asked to find the determinant of \(\left[(\mathbf{I} + \mathbf{B})^{50} - 50 \mathbf{B}\right]\). Given that \(\mathbf{B}^2 = 0\), it means that \(\mathbf{B}\) is nilpotent.

Step 2: Expand using Binomial Theorem

We can use the binomial theorem for matrices to expand \((\mathbf{I} + \mathbf{B})^{50}\). As \(\mathbf{B}^2 = 0\), all powers of \(\mathbf{B}\) that are 2 or higher will be zero. So our expansion simplifies greatly:\[(\mathbf{I} + \mathbf{B})^{50} = \mathbf{I} + 50\mathbf{B} \].

Step 3: Substitute into the Determinant

We need to find the determinant of \((\mathbf{I} + \mathbf{B})^{50} - 50 \mathbf{B}\). From the expansion, we have:\[(\mathbf{I} + \mathbf{B})^{50} = \mathbf{I} + 50\mathbf{B}\]. Now substitute into the determinant expression:\[\det\left(\mathbf{I} + 50\mathbf{B} - 50\mathbf{B}\right) = \det\left(\mathbf{I}\right)\]

Step 4: Calculate the Determinant

The determinant of the identity matrix \(\mathbf{I}\) for any size is always 1. Thus, we have:\[\det(\mathbf{I}) = 1\].

Key Concepts

Nilpotent Matrix

A nilpotent matrix is an interesting concept in linear algebra. It is a square matrix, say \(\mathbf{B}\), that satisfies the condition \(\mathbf{B}^n = 0\) for some positive integer \(n\). This implies that when the matrix is raised to a certain power, it results in the zero matrix.

For example, in the given exercise, the matrix \(\mathbf{B}\) is nilpotent because \(\mathbf{B}^2 = 0\).

• This means multiplying \(\mathbf{B}\) by itself yields the zero matrix.
• It also implies that \(\mathrm{B}\) does not have any eigenvalues other than zero.
• Nilpotent matrices are important in the study of linear transformations because they reduce the dimension of a vector space.

Understanding nilpotent matrices aids in simplifying matrix operations, especially when calculating matrix functions, like matrix exponentials.

Binomial Theorem for Matrices

The binomial theorem for matrices is a powerful tool when dealing with expressions like \( (\mathbf{I} + \mathbf{B})^n \). This is particularly useful when \(\mathbf{B}\) is a nilpotent matrix.

In the scalar world, the binomial theorem expands \((a + b)^n\) as a sum of terms involving powers of \(a\) and \(b\). In the realm of matrices, it works similarly:

• The theorem states that \((\mathbf{A} + \mathbf{B})^n\) can be expanded by using binomial coefficients, where \(\mathbf{A}\) and \(\mathbf{B}\) are matrices.
• When \(\mathbf{B}\) is nilpotent (like in our problem where \(\mathbf{B}^2 = 0\)), terms involving \(\mathbf{B}^2\) or higher powers disappear.
• Therefore, \((\mathbf{I} + \mathbf{B})^{50}\) becomes \(\mathbf{I} + 50\mathbf{B}\) when expanded.

This simplification not only reduces computational complexity but also makes it easier to handle matrix expressions in further mathematical operations.

Identity Matrix Determinant

An identity matrix, often denoted by \(\mathbf{I}\), is a special square matrix where all the elements on the main diagonal are 1, and all off-diagonal elements are 0. The identity matrix has a unique determinant property.

For any size of an identity matrix:\(\mathbf{I}_n\), the determinant is always 1. This is due to the fact that:

• The product of the diagonal elements (all ones) gives 1.
• An identity matrix does not change other matrices when used in multiplication: \(\mathbf{A} \times \mathbf{I} = \mathbf{I} \times \mathbf{A} = \mathbf{A}\).

In the context of determinants, the identity matrix serves as a baseline. In the exercise given, calculating \(\det(\mathbf{I})\) after making simplifications results in a determinant value of 1.

This concept forms a fundamental understanding of why certain matrix operations result in specific outcomes and is essential in larger determinant calculations of more complex matrices.