Evaluating the Determinant of a Matrix with Complex Entries

The problem asks us to evaluate the determinant of a 3x3 matrix whose entries are complex numbers and express the result in the form \(x + iy\). We then need to find the values of \(x\) and \(y\). This is a standard procedure for matrix evaluation involving complex entries.

Understanding the Determinant Calculation

For a 3x3 matrix \(\left| {\begin{array}{*{20}{c}} a&b&c\\ d&e&f\\ g&h&i \end{array}} \right|\), the determinant is calculated using the formula:

\(\text{Determinant} = a(ei - fh) - b(di - fg) + c(dh - eg)\)

In our given matrix, the entries are:

So, \(a = 6i\), \(b = -3i\), \(c = 1\), \(d = 4\), \(e = 3i\), \(f = -1\), \(g = 20\), \(h = 3\), \(i = i\).

Step-by-Step Determinant Evaluation

Let's calculate the determinant using the formula, paying close attention to the complex numbers:

\(\text{Determinant} = 6i \left( (3i)(i) - (-1)(3) \right) - (-3i) \left( (4)(i) - (-1)(20) \right) + 1 \left( (4)(3) - (3i)(20) \right)\)

Now, let's simplify each term:

Term 1: \(6i \left( (3i)(i) - (-1)(3) \right)\)
\(= 6i \left( 3i^2 - (-3) \right)\)
\(= 6i \left( 3(-1) + 3 \right)\) (Since \(i^2 = -1\))
\(= 6i \left( -3 + 3 \right)\)
\(= 6i (0) = 0\)

Term 2: \(- (-3i) \left( (4)(i) - (-1)(20) \right)\)
\(= 3i \left( 4i - (-20) \right)\)
\(= 3i \left( 4i + 20 \right)\)
\(= (3i)(4i) + (3i)(20)\)
\(= 12i^2 + 60i\)
\(= 12(-1) + 60i\) (Since \(i^2 = -1\))
\(= -12 + 60i\)

Term 3: \(1 \left( (4)(3) - (3i)(20) \right)\)
\(= 1 \left( 12 - 60i \right)\)
\(= 12 - 60i\)

Now, combine the terms to find the total determinant:

\(\text{Determinant} = (\text{Term 1}) + (\text{Term 2}) + (\text{Term 3})\)
\(\text{Determinant} = 0 + (-12 + 60i) + (12 - 60i)\)
\(\text{Determinant} = -12 + 60i + 12 - 60i\)
\(\text{Determinant} = (-12 + 12) + (60i - 60i)\)
\(\text{Determinant} = 0 + 0i\)
\(\text{Determinant} = 0\)

Identifying Real and Imaginary Parts

The result of the determinant evaluation is \(0\). We are given that the determinant equals \(x + iy\).

So, \(x + iy = 0\)

We can write 0 as \(0 + 0i\). Comparing the real and imaginary parts of \(x + iy = 0 + 0i\), we get:

• Real part: \(x = 0\)
• Imaginary part: \(y = 0\)

Thus, the values of x and y are 0 and 0, respectively. This matrix evaluation results in a determinant of zero.

In summary, performing the calculation of this determinant with complex entries led to a value of 0. By comparing this result to the form \(x + iy\), we determined the values of x and y.