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Chapter 2: Problem 150

If \(\left|\begin{array}{rrr}6 \mathrm{i} & -3 \mathrm{i} & 1 \\ 4 & 3 \mathrm{i} & -1 \\ 20 & 3 & \mathrm{i}\end{array}\right|=\mathrm{x}+\) iy then (a) \(\mathrm{x}=3, \mathrm{y}=1\) (b) \(x=1, y=3\) (c) \(x=0, y=3\) (a) \(\mathrm{x}=0, \mathrm{y}=0\)

Short Answer

Expert verified
The determinant of the given matrix is -54 + 30i, which does not match any of the provided answer choices. There might be a typo in the question or answer choices. Please verify the problem statement and answer choices.

Step by step solution

01

Rewrite the matrix with complex numbers in rectangular form.

We will write the given matrix with complex numbers in rectangular form for easier calculations: \(\begin{bmatrix} 0+6i & 0-3i & 1 \\ 4+0i & 0+3i & -1 \\ 20+0i & 3+0i & 0+1i \end{bmatrix}\)
02

Calculate the determinant using cofactor expansion method along the first column.

Using cofactor expansion method along the first column, we will find the determinant value: \(M = (0+6i)[(0+3i)(0+1i) - (-1)(3+0i)] - (4+0i)[(0-3i)(0+1i) - (-1)(20+0i)] + (0+0i)[Omitted]\) Here last term is omitted because it will be multiplied by zero and thus will not contribute anything to the determinant.
03

Evaluate the matrix determinant expression.

Now, we'll evaluate the matrix determinant expression: \(M = (0+6i)[(-3i + 3) + 3] - (4+0i)[(-2 - 3i) + 20]\) \(M = (0+6i)[(0 - 3i) + (0+3)] - (4+0i)[(18 - 3i)]\)
04

Perform the multiplications.

Performing the multiplications: \(M = (0+6i)(0+3 + 0-3i) - (4+0i)(18 - 3i)\) \(M = (6i)(3-3i) - 4(18 - 3i)\)
05

Multiply and simplify the expression.

We'll multiply and simplify the expression to find the final determinant: \(M = 18i - 18i^2 - 72 + 12i\) Since \(i^2 = -1\), we can simplify the expression as follows: \(M = 18i + 18 - 72 + 12i\) \(M = -54 + 30i\)
06

Compare the result with given options.

Now, let's compare the obtained result with the given answer choices: a) x = 3, y = 1 b) x = 1, y = 3 c) x = 0, y = 3 d) x = 0, y = 0 Our result, M = -54 + 30i, does not match any of the given answer choices. So, there might be a typo in the question or answer choices. Please verify the problem statement and answer choices.

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Most popular questions from this chapter

The inequality \(|z-4|<|z-2|\) represent the region given by (a) \(\operatorname{Re}(z)>0\) (b) \(\operatorname{Re}(z)<0\) (c) \(\operatorname{Re}(z)>2\) (d) \(\operatorname{Re}(z)>3\)

Let \(\alpha, \beta\) be real and \(z\) be a complex number if \(z^{2}+\alpha z+\beta=0\) has two distinct roots on the line \(\operatorname{Re}(z)=1\) then it is necessary that (a) \(\beta \in(-1,0)\) (b) \(|\beta|=1\) (c) \(\beta \in[1, \infty)\) (d) \(\beta \in(0,1)\)

If \(z=\cos \theta+i \sin \theta\) then \(\arg \left(z^{2}+\underline{z}\right)=\) (a) \((3 \theta / 2)\) (b) \(\theta\) (c) \((\theta / 2)\) (d) \(3 \theta\)

It \(\mathrm{z}^{2}+\mathrm{z}+1=0\) where \(\mathrm{z}\) is a complex number, then the value of \([z+(1 / z)]^{2}+\left[z^{2}+\left(1 / z^{2}\right)\right]^{2}+\left[z^{3}+\left(1 / z^{3}\right)\right]^{2}+\ldots\) \(+\left[z^{6}+\left(1 / z^{6}\right)\right]^{2}\) is (a) 18 (b) 54 (c) 6 (d) 12

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