Problem 28 - Algebra for JEE ADVANCED

Question

If \(A\) is a nilpotent matrix of index 2 , then for any positive integer \(n, A(I+A)^{n}\) is equal to a. \(A^{-1}\) b. \(A\) c. \(A^{n}\) d. \(I_{n}\)

Step-by-Step Solution

Answer

The answer is b: \(A\).

Step 1: Understand the properties of nilpotent matrices

A nilpotent matrix is a square matrix such that for some positive integer k, the matrix raised to the power of k is the zero matrix. Since given that matrix A is nilpotent of index 2, we have the property that \(A^2 = 0\).

Step 2: Simplify the expression \((I+A)^n\)

Since we know \(A^2 = 0\), when expanding \((I+A)^n\) using the binomial theorem, only terms up to \(A\) are non-zero because any power higher than \(A^1\) involving \(A\) will contribute to zero. Therefore, \((I+A)^n = I + nA\).

Step 3: Calculate \(A(I+A)^n\)

Using the expansion \((I+A)^n = I + nA\), we multiply by A to get:\(A(I + nA) = AI + AnA = A + nA^2\).Since \(A^2 = 0\), this simplifies to:\(A(I+A)^n = A + n \cdot 0 = A\).

Step 4: Conclude the answer

From the calculations, we determined that \(A(I+A)^n = A\). Therefore, the correct answer is option (b) \(A\).

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with the study of matrices and their operations. Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. In the context of algebra:

• Addition and subtraction of matrices involve combining corresponding elements from each matrix.
• Matrix multiplication is more complex; it requires summing the products of elements across the rows of one matrix and columns of another.

The identity matrix, denoted as **I**, plays a crucial role in matrix algebra. It functions similar to the number 1 in ordinary arithmetic, such that when any matrix is multiplied by the identity matrix, it remains unchanged (\(AI = IA = A\)). Understanding the properties of matrix operations is fundamental to solving problems in matrix algebra effectively.

Binomial Theorem

The binomial theorem is a powerful tool in algebra that provides a formula to expand expressions raised to a power. For a binomial expression \((x + y)^n\), it can be expanded as:

• \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}\),

where \(\binom{n}{k}\) is a binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\). When applied to matrices, the theorem helps in expanding expressions like \((I + A)^n\). In cases where matrix \(A\) is nilpotent with \(A^2=0\), terms with powers of \(A^2\) or higher become zero. This significantly simplifies expressions and reveals results like \((I + A)^n = I + nA\), a critical step in solving related matrix problems.

Matrix Index

The concept of a matrix index is especially significant in defining nilpotent matrices. A nilpotent matrix \(A\) has an index \(k\) if \(A^k = 0\) but \(A^{k-1} eq 0\). The smallest \(k\) for which this is true is called the index of the matrix. In our problem, \(A\) is nilpotent with an index of 2, meaning \(A^2 = 0\) but \(A eq 0\). This property drastically reduces the complexity in calculations involving \(A\), as it implies any higher power will result in a zero matrix. Understanding the index helps predict the behavior of a matrix when involved in algebraic expressions.

Zero Matrix

A zero matrix is a matrix where all elements are zero. Denoted by \(0\), it is the additive identity in matrix algebra. For any matrix \(A\), when you add the zero matrix, you retain the original matrix:

• \(A + 0 = A\)

In the context of multiplication, for a matrix \(A\), multiplying it by a zero matrix results in another zero matrix of appropriate size:

• \(A \, 0 = 0\)

When dealing with nilpotent matrices like in our original exercise, where \(A^2 = 0\), the zero matrix represents the result of squaring the matrix, demonstrating its transformative properties. Recognizing this concept is key in understanding and predicting results in various matrix operations, as seen in algebraic and theoretical problems alike.