Problem 30 - Precalculus

Question

Let \(z=a+b i\) and \(w=c+d i\) be arbitrary complex numbers. Associate \(z\) and \(w\) with the matrices $$ Z=\left[\begin{array}{rr} a & b \\ -b & a \end{array}\right] \text { and } W=\left[\begin{array}{rr} c & d \\ -d & c \end{array}\right] $$ Show that complex number addition, subtraction and multiplication are mirrored by the associated matrix arithmetic. That is, show that \(Z+W, Z-W\) and \(Z W\) produce matrices which can be associated with the complex numbers \(z+w, z-w\) and \(z w,\) respectively.

Step-by-Step Solution

Answer

Matrix addition, subtraction, and multiplication correspond to complex number operations.

Step 1: Matrix Association of Addition

The complex number addition \( z + w \) is given by \((a + bi) + (c + di) = (a+c) + (b+d)i \). The matrix representation of this number is:\\[ Z + W = \begin{bmatrix} a & b \ -b & a \end{bmatrix} + \begin{bmatrix} c & d \ -d & c \end{bmatrix} \]\Adding the matrices component-wise, we get:\\[ Z + W = \begin{bmatrix} a+c & b+d \ -(b+d) & a+c \end{bmatrix} \]\This matrix represents the complex number \((a+c) + (b+d)i\) which corresponds to \(z+w\).

Step 2: Matrix Association of Subtraction

The complex number subtraction \( z - w \) is given by \((a + bi) - (c + di) = (a-c) + (b-d)i \). Its matrix representation is:\\[ Z - W = \begin{bmatrix} a & b \ -b & a \end{bmatrix} - \begin{bmatrix} c & d \ -d & c \end{bmatrix} \]\Performing matrix subtraction component-wise results in:\\[ Z - W = \begin{bmatrix} a-c & b-d \ -(b-d) & a-c \end{bmatrix} \]\This matrix corresponds to the complex number \((a-c) + (b-d)i\), which matches \(z-w\).

Step 3: Matrix Association of Multiplication

For multiplication, the complex number \( z \cdot w \) is \((a+bi)(c+di) = (ac-bd) + (ad+bc)i \). The matrix product is:\\[ZW = \begin{bmatrix} a & b \ -b & a \end{bmatrix} \times \begin{bmatrix} c & d \ -d & c \end{bmatrix} = \begin{bmatrix} ac-bd & ad+bc \ -(ad+bc) & ac-bd \end{bmatrix}\]\This resulting matrix represents the complex number \((ac-bd) + (ad+bc)i\), which corresponds to the multiplication \(z \cdot w\).

Step 4: Conclusion

By showing the results of matrix operations \(Z+W\), \(Z-W\), and \(ZW\) are representable as complex numbers \(z+w\), \(z-w\), and \(z\cdot w\) respectively, we have established that the operations on these matrices reflect the operations on their associated complex numbers.

Key Concepts

Matrix Representation

When dealing with complex numbers, associating them with matrices can provide a more visual and structured approach, especially for operations like addition, subtraction, and multiplication. If we have complex numbers expressed as \( z = a + bi \) and \( w = c + di \), they can be translated into matrices:
- For \( z \), the matrix is \( Z = \begin{bmatrix} a & b \ -b & a \end{bmatrix} \). - Similarly, for \( w \), it becomes \( W = \begin{bmatrix} c & d \ -d & c \end{bmatrix} \).
These matrices help in straightforward visualization of how complex number operations work. The entries in these matrices are chosen so that the addition, subtraction, and multiplication reflect operations on the original complex numbers. This matrix representation maintains the properties needed to correctly perform and simplify these operations, as matrix arithmetic parallels the properties of complex numbers.

Complex Addition

Complex addition involves combining the real parts and the imaginary parts of two complex numbers. For example, if you have \( z = a + bi \) and \( w = c + di \), then the addition \( z + w \) results in \( (a+c) + (b+d)i \).
Translated to matrices, this operation is also straightforward. Using our given matrices \( Z \) and \( W \):
- Compute the sum by adding corresponding elements: \[ Z + W = \begin{bmatrix} a & b \ -b & a \end{bmatrix} + \begin{bmatrix} c & d \ -d & c \end{bmatrix} = \begin{bmatrix} a+c & b+d \ -(b+d) & a+c \end{bmatrix} \]
This matrix representation directly corresponds to the complex number \((a+c) + (b+d)i\), ensuring the parallelism between matrix and complex arithmetic.

Complex Subtraction

In complex subtraction, we simply subtract the corresponding real and imaginary parts of the numbers involved. For complexity numbers \( z = a + bi \) and \( w = c + di \), the subtraction \( z - w \) yields \( (a-c) + (b-d)i \).
Matrix subtraction follows a similar rule, making use of the matrix representations \( Z \) and \( W \):
- Perform element-wise subtraction: \[ Z - W = \begin{bmatrix} a & b \ -b & a \end{bmatrix} - \begin{bmatrix} c & d \ -d & c \end{bmatrix} = \begin{bmatrix} a-c & b-d \ -(b-d) & a-c \end{bmatrix} \]
This mirrors the complex number \((a-c) + (b-d)i\), matching precisely the approach used for complex numbers.

Complex Multiplication

Complex multiplication can be a bit more intricate, involving the distribution of terms and combining them according to imaginary unit rules. For two complex numbers \( z = a + bi \) and \( w = c + di \), the multiplication \( z \cdot w \) results in \( (ac-bd) + (ad+bc)i \).
In matrix form, the operation involves multiplying the matrices \( Z \) and \( W \):
- Apply matrix multiplication rules: \[ ZW = \begin{bmatrix} a & b \ -b & a \end{bmatrix} \times \begin{bmatrix} c & d \ -d & c \end{bmatrix} = \begin{bmatrix} ac-bd & ad+bc \ -(ad+bc) & ac-bd \end{bmatrix} \]
The resulting matrix encodes the same complex number result \((ac-bd) + (ad+bc)i\), validating the consistency and unity between matrix and complex number multiplications. This approach underscores the concept of seamlessly integrating matrix operations with complex arithmetic.