Problem 92 - Precalculus with Limits

Question

Let \(i = \sqrt{-1}\) and let \(A = \left[ \begin{array}{r} \)i\( & 0 \\ 0 & \)i\(\end{array} \right]\), \(B = \left[ \begin{array}{r} 0 & -i \\ i & 0 \end{array} \right]\). (a) Find \(A^2\), \(A^3\), and \(A^4\). Identify any similarities with \(i^2\), \(i^3\) and \(i^4\). (b) Find and identify \(B^2\).

Step-by-Step Solution

Answer

The powers of the matrices follow the pattern of the powers of \(i\). Specifically, \(A^2 = i^2 = -1, A^3 = i^3 = -i, A^4 = i^4 = 1\), and \(B^2 = i^2 = -1\).

Step 1: Compute powers of \(i\)

To understand how the powers of \(i\) behave, calculate: \(i^1 = i\),\(i^2 = -1\), because \(i\) is defined as \(\sqrt{-1}\),\(i^3 = -i\), which is \(i^2 \cdot i = -1 \cdot i = -i\),\(i^4 = 1\), because \(i^4 = i^2 \cdot i^2 = -1 \cdot -1 = 1\).Note that these powers cycle in a pattern: \(i, -1, -i, 1,...\)

Step 2: Compute powers of Matrix A

Now calculate the powers of matrix A. Remember that matrix multiplication is not commutative, and the order of multiplication matters.Firstly, \(A^2 = A \cdot A =\left[ \begin{array}{r} i & 0 \ 0 & i \end{array} \right] \cdot \left[ \begin{array}{r} i & 0 \ 0 & i \end{array} \right] = \left[ \begin{array}{r} -1 & 0 \ 0 & -1 \end{array} \right]\).This matches the pattern of \(i^2 = -1\).Next, \(A^3 = A \cdot A^2 = \left[ \begin{array}{r} i & 0 \ 0 & i \end{array} \right] \cdot \left[ \begin{array}{r} -1 & 0 \ 0 & -1 \end{array} \right] = \left[ \begin{array}{r} -i & 0 \ 0 & -i \end{array} \right]\),which resembles the pattern of \(i^3 = -i\).Lastly, \(A^4 = A \cdot A^3 = \left[ \begin{array}{r} i & 0 \ 0 & i \end{array} \right] \cdot \left[ \begin{array}{r} -i & 0 \ 0 & -i \end{array} \right] = \left[ \begin{array}{r} 1 & 0 \ 0 & 1 \end{array} \right]\).This is identical to \(i^4 = 1\). Therefore, the powers of matrix A follow the same pattern as the powers of \(i\).

Step 3: Compute powers of Matrix B

Compute the square of matrix B:\(B^2 = B \cdot B = \left[ \begin{array}{r} 0 & -i \ i & 0 \end{array} \right] \cdot \left[ \begin{array}{r} 0 & -i \ i & 0 \end{array} \right] = \left[ \begin{array}{r} -1 & 0 \ 0 & -1 \end{array} \right]\). This is the same as \(i^2 = -1\)

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. Unlike regular multiplication with numbers, matrix multiplication involves combining the rows of the first matrix with the columns of the second matrix. An important thing to note is that matrix multiplication is not commutative. This means if you have two matrices, say matrix A and matrix B, the product \(AB\) might not be the same as \(BA\).

• To multiply two matrices, you have to have compatible dimensions. This means if you have a matrix A of size \(m \times n\), you can only multiply it by a matrix B of size \(n \times p\). The resulting matrix will then be \(m \times p\).
• Also important is to follow the order: you multiply the elements from corresponding rows of the first matrix by the columns of the second matrix and then sum those products.

Understanding matrix multiplication helps us navigate through complex numbers structured in matrix form. For instance, the lesson breaks down multiplying matrices A and B, both involving the imaginary unit \(i\). In these exercises, keeping track of the imaginary units is key to achieving correct results.

Powers of i

The imaginary unit \(i\), defined as \(\sqrt{-1}\), is fundamental in working with complex numbers. Understanding the powers of \(i\) is vital as they follow a repetitive cycle which simplifies calculations when dealing with complex expressions.

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)
• After every four powers, the cycle repeats: \(i, -1, -i, 1, i, -1, -i, 1, \ldots\)

When you are calculating the powers of matrices involving \(i\), as seen in matrix A calculations, you observe similarities with these power cycles. Matrices \(A^2, A^3,\) and \(A^4\) follow the same cycling pattern observed in individual powers of \(i\), demonstrating the powerful coherence between these mathematical structures.

Imaginary Unit

The imaginary unit, \(i\), serves as the basis of complex numbers. It is defined with the unique property \(i = \sqrt{-1}\), meaning that \(i^2 = -1\). This characteristic redefines how we handle numbers, particularly those associated with negative square roots, which real numbers can't manage.

• The inclusion of \(i\) expands the real number line into a complex plane which includes all possible linear combinations of the form \(a + bi\), where \(a\) and \(b\) are real numbers.
• It also embodies a function that allows for the representation of cyclic behaviors as shown by the periodic nature of its powers.
• Using \(i\) in matrices not only enhances computational capabilities but also provides new perspectives in solving problems that have no solutions in the realm of real numbers.

In exercises involving matrices \(A\) and \(B\), \(i\) plays a crucial role. It not only allows these matrices to represent more complex structures but also unveils the connection between imaginary numbers and matrix algebra, showcasing their applications beyond just basic arithmetic.