Problem 14 - Differential Equations and Linear Algebra

Question

An \(n \times n\) matrix \(A\) is called nilpotent if \(A^{p}=0\) for some positive integer \(p .\) Show that the given matrix is nilpotent. $$A=\left[\begin{array}{rr}3 & 9 \\ -1 & -3\end{array}\right]$$

Step-by-Step Solution

Answer

The given matrix A is nilpotent because when raised to the power of 2, it results in a zero matrix: \[A^2=\left[\begin{array}{rr}3 & 9 \\\ -1 & -3\end{array}\right]^2=\left[\begin{array}{rr}0 & 0 \\\ 0 & 0\end{array}\right]\]

Step 1: Compute Matrix Powers

We start by computing the powers of matrix A, starting from A^2, and continue until we get a zero matrix or a pattern is found. Calculate \(A^2\): \(A^2 = AA\)

Step 2: Compute Matrix Multiplication

Compute the product of matrix A by itself: \[A^2=\left[\begin{array}{rr}3 & 9 \\\ -1 & -3\end{array}\right]\left[\begin{array}{rr}3 & 9 \\\ -1 & -3\end{array}\right]=\left[\begin{array}{rr}(3 \cdot 3) + (9 \cdot (-1)) & (3 \cdot 9) + (9 \cdot (-3)) \\\ (-1 \cdot 3) + (-3 \cdot (-1)) & (-1 \cdot 9) + (-3 \cdot (-3))\end{array}\right]\]

Step 3: Simplify Result

Simplify the components of \(A^2\): \[A^2=\left[\begin{array}{rr}0 & 0 \\\ 0 & 0\end{array}\right] \]

Step 4: Verify Nilpotent Condition

Since \(A^2\) is a zero matrix, it meets the condition for a nilpotent matrix, as \(A^p=0\) with \(p=2\). Thus, the given matrix A is nilpotent.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves multiplying two matrices together to produce a third matrix. For two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second one.
Matrix multiplication is not merely multiplying the corresponding elements but involves a more complex process:

• The entry in the first row and first column of the product is the sum of the products of the entries in the first row of the first matrix and the first column of the second matrix.
• Continue this process for each position in the resultant matrix.

In our example, when multiplying the matrix \( A \) with itself, we performed these calculations for each entry to ultimately derive \( A^2 \). Understanding matrix multiplication is crucial for exploring deeper concepts like matrix powers and nilpotent matrices.

Zero Matrix

A zero matrix is a matrix in which every element is zero. It serves as the additive identity in matrix operations, much like zero in elementary arithmetic.
When we multiply any matrix by a zero matrix, the result is also a zero matrix. This characteristic makes zero matrices pivotal when discussing nilpotent matrices. It signifies that repeated matrix operations, like taking powers of the matrix, may simplify to a matrix full of zeroes.
For matrix \( A \) in our exercise, finding that \( A^2 \) is a zero matrix means it plays a key role in confirming \( A \)'s nilpotent nature without further calculations. It's the end goal when we seek to prove nilpotent conditions.

Matrix Powers

Matrix powers refer to the result of repeatedly multiplying a square matrix by itself. For any square matrix \( A \), \( A^2 \) is the matrix multiplied by itself once, \( A^3 \) is \( A^2 \) multiplied by \( A \) again, and so on.
Understanding matrix powers is essential in analyzing properties of matrices, such as whether they are nilpotent. With each multiplication, patterns might emerge that simplify the process.

• In our example, we quickly reach a zero matrix upon computing \( A^2 \).
• This not only saves further computations but definitively indicates that no higher powers needs to be checked because the nilpotent condition is fulfilled.

This highlights how matrix powers help identify matrix characteristics up efficiently, especially when paired with zero matrices.

Nilpotent Condition

The nilpotent condition is a special property of square matrices. A matrix \( A \) is defined as nilpotent if there exists a positive integer \( p \) such that \( A^p = 0 \), where \( 0 \) represents the zero matrix.
This property is important in theoretical studies and applications of matrices. It implies the matrix reaches a state through repeated multiplication where all its information is nulled.
In our given example, matrix \( A \) has been shown to satisfy the nilpotent condition with \( p=2 \), as \( A^2 \) is already a zero matrix. This reinforces how identifying nilpotent matrices can be straightforward once you prove the power condition.

• Nilpotent matrices have interesting mathematical implications, especially in systems that leverage iterated processes like certain differential equations or iterative algorithms.

Understanding nilpotency simplifies complex matrix behaviors in computational tasks and theoretical analyses.