Question

If \(z\) is a complex number and \(a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}\) are all real, then $$ \left|\begin{array}{lll} a_{1} z+b_{1} z & a_{2} z+b_{2} z & a_{3} z+b_{3} z \\ b_{1} z+a_{1} z & b_{2} z+a_{2} z & b_{3} z+a_{3} z \\ b_{1} z+a_{1} & b_{2} z+a_{2} & b_{3} z+a_{3} \end{array}\right|=\ldots $$ (a) \(|\underline{z}|^{2}\) (b) \(\left(a_{1} a_{2} a_{3}+b_{1} b_{2} b_{3}\right)^{2}|z|^{2}\) (c) c (d) 0

Short answer

The short answer is: (d) 0

Step-by-step solution

Identify the matrix

First, identify the given matrix: \[ A = \left(\begin{array}{lll} a_{1} z+b_{1} z & a_{2} z+b_{2} z & a_{3} z+b_{3} z \\\ b_{1} z+a_{1} z & b_{2} z+a_{2} z & b_{3} z+a_{3} z \\\ b_{1} z+a_{1} & b_{2} z+a_{2} & b_{3} z+a_{3} \end{array}\right) \]

Calculate the determinant

To find the determinant of a \(3 \times 3\) matrix, we will use the rule: \(|A| = a(ei - fh) - b(di - fg) + c(dh - eg) \) where \(a, b, c, d, e, f, g, h, i\) are the matrix elements. Let's calculate it for our matrix: \(|A| = (a_1z + b_1z)((b_2z + a_2z)(b_3z + a_3) - (b_3z + a_3z)(b_2z + a_2)) - (a_2z + b_2z)((b_1z + a_1z)(b_3z + a_3) - (b_3z + a_3z)(b_1z + a_1)) + (a_3z + b_3z)((b_1z + a_1z)(b_2z + a_2) - (b_2z + a_2z)(b_1z + a_1))\)

Simplify the expression

Now, let's simplify the above complex expression. Notice that each of the terms within parenthesis have a common factor of \(z\). We can factor this out: \(|A| = z^3\left(\left(a_1 + b_1\right)((b_2 + a_2)(b_3 + a_3) - (b_3 + a_3)(b_2 + a_2)) - \left(a_2 + b_2\right)((b_1 + a_1)(b_3 + a_3) - (b_3 + a_3)(b_1 + a_1)) + \left(a_3 + b_3\right)((b_1 + a_1)(b_2 + a_2) - (b_2 + a_2)(b_1 + a_1))\right)\) Upon further simplification, we'll notice that each of the terms within the parentheses in the determinant cancel out: \(|A| = z^3(0) = 0\)

Choose the correct answer

The determinant of the given matrix simplifies to \(0\). Therefore, the correct answer is: (d) 0

Key concepts

Determinant of a 3x3 Matrix

Calculating the determinant of a 3x3 matrix can verify its invertibility and properties. The formula for the determinant is a bit complex but follows a clear pattern:

• Choose an element from a row, let's say the first row.
• Expand the determinant using cofactor expansion across this row.
• Essentially, the determinant is computed by breaking the matrix into smaller, more manageable pieces.

In our original exercise, each element of the matrix involved complex expressions based on the variable \(z\) and real numbers. It can become cumbersome, but the principle remains the same: decompose the matrix into smaller determinants. In simpler terms, you are calculating the sums and differences of the products of the elements across the matrix by following the rule of: \[|A| = a(ei - fh) - b(di - fg) + c(dh - eg)\] The majority of operations involve multiplication and addition, so remember your priority is to preserve these operations' integrity to ensure correct outcomes. By methodically breaking down, you can manage even intricate expressions.

Complex Numbers

Complex numbers have both a real and an imaginary part. They are written in the form \(z = a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part, with \(i\) representing the square root of \(-1\).
These numbers are crucial when dealing with matrices that involve imaginary components. The provided exercise consisted of expressions where \(z\), a complex number, multiplies with real elements. This can initially seem intimidating if you're used to matrices with pure real numbers.

• Adding and multiplying complex numbers follow the distributive law, similar to real numbers but include combining like terms of the real and imaginary components.
• Conjugation is important, especially when simplifying expressions within matrices. The complex conjugate of \(z\) is \(\overline{z} = a - bi\).

Recognizing patterns in complex numbers can help ensure accuracy when solving matrix problems, especially when encountering factors like \( |z|^2 = z\overline{z} \), it's vital to understand how these terms interact to simplify your results in problems involving determinants.

Matrix Simplification

Simplifying matrices involves reducing them to a form where patterns or repetitive calculations are more evident, helping ease complex operations such as finding determinants. With the exercise at hand, using operations like factoring out common terms across matrix elements simplifies daunting expressions considerably.
Here are some key techniques for matrix simplification:

• Factor common elements from rows or columns to reduce the complexity of the original matrix expressions.
• Use matrix properties, like canceling elements, especially invaluable when complex numbers are involved, as in our example.
• Prioritize operations that allow terms to cancel. Look out for opportunities to nullify terms, as shown in the exercise, where simplification led to each expression in the determinant nullifying to zero.

Understanding these simplification methods significantly improves efficiency in calculations and avoids being overwhelmed by the multitude of terms present in complex matrices.