Question

If $\begin{vmatrix} {6i}&{ - 3i}&1 \\ 4&{3i}&{ - 1} \\ {20}&3&i \end{vmatrix} = x + iy$ then
A. $x = 3,y = 1$
B. $x = 3,y = 0$
C. $x = 0,y = 1$
D. $x = 0,y = 0$

Answer 1

Hint: In this question, for determining the values of $x$ and $y$, we have to simplify the given equation and then compare the real parts and imaginary parts of real numbers. We need to use the sum of product rules to determine the value of a determinant.

Complete step by step solution:
We know that $\begin{vmatrix} {6i}&{ - 3i}&1 \\ 4&{3i}&{ - 1} \\ {20}&3&i \end{vmatrix} = x + iy$
Let us simplify the determinant first.
$\begin{vmatrix} {6i}&{ - 3i}&1 \\ 4&{3i}&{ - 1} \\ {20}&3&i \end{vmatrix} = x + iy$
$\Rightarrow 6i\left( {\left( {3i \times i} \right) - \left( { - 1 \times 3} \right)} \right) - \left( { - 3i} \right)\left( {\left( {4 \times i} \right) - \left( { - 1 \times 20} \right)} \right) + 1\left( {4 \times 3} \right) - \left( {20 \times 3i} \right) $
Thus, we get
$ \Rightarrow 6i\left( { - 3 + 3} \right) + 3i\left( {4i + 20} \right) + 1\left( {12 - 60i} \right) = x + iy$
By simplifying further, we get
$ \Rightarrow - 18i + 18i + 12{i^2} + 60i + 12 - 60i = x + iy$
Put ${i^2} = - 1$
$ \Rightarrow 12\left( { - 1} \right) + 12 = x + iy \\ \Rightarrow 0 + 0i = x + iy $
Thus, by comparing real parts and imaginary parts, we get
$x = 0; y = 0$
Thus, both the variables have zero value.

Option ‘D’ is correct

Additional Information: The determinant is related to a square matrix and defined as a scalar value which is associated with the square matrix. That means, it is a square array of numbers which indicates a particular sum of products. Also, the complex numbers are the combinations of two parts such as a real number and an imaginary number. Mathematically, it can be expressed as where, are real numbers and is an imaginary number which is called “iota”. Also, A complex number can be shown as a pair of integers creating a vector on an Argand diagram, which represents the complex plane.

Note: Many students make mistakes in calculating the determinant and further part. This is the only way through which we can solve the example in the simplest way. Many students have been confused with the concept of complex numbers and make mistakes in comparison of real and imaginary parts of a complex number.