Problem 17 - Advanced Engineering Mathematics

Question

A nonzero \(n \times n\) matrix \(\mathbf{A}\) is said to be nilpotent of index \(m\) if \(m\) is the smallest positive integer for which \(\mathbf{A}^{m}=\mathbf{0}\). Which of the following matrices are nilpotent? If nilpotent, what is its index? (a) \(\left(\begin{array}{rr}1 & 0 \\ -1 & 0\end{array}\right)\) (b) \(\left(\begin{array}{rr}2 & 2 \\ -2 & -2\end{array}\right)\) (c) \(\left(\begin{array}{lll}0 & 0 & 0 \\ 1 & 0 & 0 \\ 2 & 3 & 0\end{array}\right)\) (d) \(\left(\begin{array}{lll}0 & 0 & 5 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)\) (e) \(\left(\begin{array}{rrrr}0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ -1 & 0 & 0 & 0\end{array}\right)\) (f) \(\left(\begin{array}{llll}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 3 & 1 & 0 & 0 \\ 2 & 2 & 1 & 0\end{array}\right)\)

Step-by-Step Solution

Answer

Matrices (b), (c), (d), (e), and (f) are nilpotent with indices 2, 3, 2, 3, and 4, respectively.

Step 1: Check Matrix (a) for Nilpotency

Calculate \(\mathbf{A}^2\) for the matrix \( \mathbf{A} = \begin{pmatrix} 1 & 0 \ -1 & 0 \end{pmatrix} \).\[\mathbf{A}^2 = \begin{pmatrix} 1 & 0 \ -1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \ -1 & 0 \end{pmatrix} = \begin{pmatrix} 1 \cdot 1 + 0 \cdot -1 & 1 \cdot 0 + 0 \cdot 0 \-1 \cdot 1 + 0 \cdot -1 & -1 \cdot 0 + 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \ -1 & 0 \end{pmatrix} \]Since \(\mathbf{A}^2 eq \mathbf{0}\), the matrix is not nilpotent.

Step 2: Check Matrix (b) for Nilpotency

Calculate \(\mathbf{A}^2\) for the matrix \( \mathbf{A} = \begin{pmatrix} 2 & 2 \ -2 & -2 \end{pmatrix} \).\[\mathbf{A}^2 = \begin{pmatrix} 2 & 2 \ -2 & -2 \end{pmatrix} \begin{pmatrix} 2 & 2 \ -2 & -2 \end{pmatrix} = \begin{pmatrix} 2 \cdot 2 + 2 \cdot -2 & 2 \cdot 2 + 2 \cdot -2 \-2 \cdot 2 + -2 \cdot -2 & -2 \cdot 2 + -2 \cdot -2 \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} \]So, \(\mathbf{A}^2 = \mathbf{0}\), thus matrix (b) is nilpotent of index 2.

Step 3: Check Matrix (c) for Nilpotency

Calculate powers \(\mathbf{A}^2\) and \(\mathbf{A}^3\) for matrix \( \mathbf{A} = \begin{pmatrix} 0 & 0 & 0 \ 1 & 0 & 0 \ 2 & 3 & 0 \end{pmatrix}\).\[\mathbf{A}^2 = \begin{pmatrix} 0 & 0 & 0 \ 1 & 0 & 0 \ 2 & 3 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 \ 1 & 0 & 0 \ 2 & 3 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 3 & 0 & 0 \end{pmatrix}\]\[\mathbf{A}^3 = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 3 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 \ 1 & 0 & 0 \ 2 & 3 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}\]Matrix (c) is nilpotent of index 3 since \(\mathbf{A}^3 = \mathbf{0}\) and \(\mathbf{A}^2 eq \mathbf{0}\).

Step 4: Check Matrix (d) for Nilpotency

Calculate powers \(\mathbf{A}^2\) and \(\mathbf{A}^3\) for matrix \(\mathbf{A} = \begin{pmatrix} 0 & 0 & 5 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}\).\[\mathbf{A}^2 = \begin{pmatrix} 0 & 0 & 5 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 5 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}\]Matrix (d) is nilpotent of index 2 since \(\mathbf{A}^2 = \mathbf{0}\).

Step 5: Check Matrix (e) for Nilpotency

Calculate powers \(\mathbf{A}^2\), \(\mathbf{A}^3\), and \(\mathbf{A}^4\) for matrix \( \mathbf{A} = \begin{pmatrix} 0 & 0 & 0 & 0 \ -1 & 0 & 0 & 1 \ 0 & 1 & 0 & 1 \ -1 & 0 & 0 & 0 \end{pmatrix} \).\[\mathbf{A}^2 = \begin{pmatrix} 0 & 0 & 0 & 0 \ -1 & 0 & 0 & 1 \ 0 & 1 & 0 & 1 \ -1 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 & 0 \ -1 & 0 & 0 & 1 \ 0 & 1 & 0 & 1 \ -1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ -1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{pmatrix} \]\[\mathbf{A}^3 = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ -1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 & 0 \ -1 & 0 & 0 & 1 \ 0 & 1 & 0 & 1 \ -1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{pmatrix} \]Matrix (e) is nilpotent of index 3 since \(\mathbf{A}^3 = \mathbf{0}\) and \(\mathbf{A}^2 eq \mathbf{0}\).

Step 6: Check Matrix (f) for Nilpotency

Check powers \(\mathbf{A}^2\), \(\mathbf{A}^3\), and \(\mathbf{A}^4\) for matrix \(\mathbf{A} = \begin{pmatrix} 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 3 & 1 & 0 & 0 \ 2 & 2 & 1 & 0 \end{pmatrix}\).\[\mathbf{A}^2 = \begin{pmatrix} 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 3 & 1 & 0 & 0 \ 2 & 2 & 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 3 & 1 & 0 & 0 \ 2 & 2 & 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 5 & 1 & 0 & 0 \end{pmatrix} \]\[\mathbf{A}^3 = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 5 & 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 3 & 1 & 0 & 0 \ 2 & 2 & 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \end{pmatrix} \]\[\mathbf{A}^4 = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 3 & 1 & 0 & 0 \ 2 & 2 & 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{pmatrix} \]Matrix (f) is nilpotent of index 4 since \(\mathbf{A}^4 = \mathbf{0}\) and \(\mathbf{A}^3 eq \mathbf{0} \).

Key Concepts

Matrix Index

The matrix index is a crucial concept when discussing nilpotent matrices. A matrix is said to be nilpotent if there is a smallest positive integer, called the index, at which the power of the matrix equals the zero matrix. Specifically, for a matrix \( \mathbf{A} \), if the smallest integer \( m \) for which \( \mathbf{A}^m = \mathbf{0} \) holds true, then \( m \) is the index of nilpotency. This means that as you progressively multiply \( \mathbf{A} \) by itself, it eventually becomes the zero matrix. If a matrix is nilpotent of index \( m \), it suggests that repeated applications (\( m \) times) of its transformation result in annihilating any vector it acts upon. Understand the index by exploring powers of matrices and focusing on when they result in zeros throughout the entire matrix.

Matrix Powers

Matrix powers involve multiplying a matrix by itself a certain number of times. Consider calculating \( \mathbf{A}^2 \): this is simply the matrix \( \mathbf{A} \) multiplied by itself. Similarly, \( \mathbf{A}^3 \) is the result of multiplying \( \mathbf{A}^2 \) by \( \mathbf{A} \) once more. For nilpotent matrices, this process is especially important because it's how we determine the index of nilpotency. Start by calculating powers like \( \mathbf{A}^2 \), continue to \( \mathbf{A}^3 \), and so on until you reach \( \mathbf{A}^m = \mathbf{0} \). This process stops when all resulting elements in the matrix become zero, indicating that we've reached the power at which the matrix ceases to change any further.

Nilpotency Check

Checking for nilpotency involves determining if, and when, a matrix becomes the zero matrix through successive multiplications. You begin by calculating the initial powers, such as \( \mathbf{A}^2 \), for a given matrix \( \mathbf{A} \). If \( \mathbf{A}^2 eq \mathbf{0} \), continue to \( \mathbf{A}^3 \), \( \mathbf{A}^4 \), etc., until you find the first occurrence where \( \mathbf{A}^m = \mathbf{0} \). This is important because identifying when a matrix becomes nilpotent can reveal significant properties about the linear transformations it represents. For each matrix, the procedure includes:

• Calculate each power step by step.
• Check when the resulting matrix becomes zero in its entirety.

Once you find this power, you've successfully completed the nilpotency check.

Matrix Multiplication

Matrix multiplication is the operation used to calculate powers of matrices. Its consistency and correctness are vital to establish results like nilpotency. When multiplying two matrices, align the rows of the first matrix with the columns of the second. Calculate each entry by taking the dot product of the corresponding row and column. For example, in calculating \( \mathbf{A}^2 \) for a 2x2 matrix:

• Multiply the first row of \( \mathbf{A} \) by the first column of the subsequent matrix.
• Carry on with all rows and columns until each new entry is filled.

Ensure that the dimensions agree for valid multiplication. Mastering this operation aids not just in finding matrix powers, but in many other areas of matrix algebra.

Matrix Algebra

Matrix algebra comprises various operations, including addition, scalar multiplication, and particularly multiplication between matrices which we've extensively discussed. It provides the framework needed to analyze matrices' properties, such as nilpotency. Understanding matrix algebra helps to simplify and solve diverse problems in mathematics and engineering.

• Matrix addition and subtraction use element-wise operations.
• Scalar multiplication involves multiplying each matrix element by a scalar value.
• Most importantly, mastering the rules of matrix multiplication is essential, as it propels calculations like finding powers and subsequently, checking for nilpotency.

By exploring these various aspects, you can tackle problems involving transformations, systems of equations, and vector spaces with efficiency.