Complex Numbers

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Questions and Answers

In the complex number $z = a + bi$, what does the real part 'a' represent graphically?

The horizontal axis on the complex plane. (correct)
The vertical axis on the complex plane.
The magnitude of the complex number.
The angle with respect to the origin.

If $j$ is the imaginary unit, what is the result of $j^3$?

-1
1
-j (correct)
j

Which of the following is the correct representation of the magnitude $r$ in the polar form of a complex number $z = a + jb$?

$r = \sqrt{a + b}$
$r = a^2 + b^2$
$r = \sqrt{a^2 + b^2}$ (correct)
$r = a + b$

Which of the following equations correctly represents the conversion from Cartesian form $z = a+jb$ to exponential form?

$z = re^{j\theta}$, where $\theta = tan^{-1}(\frac{b}{a})$ (C) Signup and view all the answers

Given two complex numbers $z_1 = a + jb$ and $z_2 = c + jd$, what is the result of $z_1 + z_2$?

$(a + c) + j(b + d)$ (C) Signup and view all the answers

Given complex numbers $z_1 = r_1∠θ_1$ and $z_2 = r_2∠θ_2$, what is $z_1 * z_2$?

$(r_1 * r_2)∠(θ_1 + θ_2)$ (B) Signup and view all the answers

What is the complex conjugate of the complex number $z = a + jb$?

a - jb (D) Signup and view all the answers

Given a complex number $z = a + jb$, how is $log(z)$ calculated?

$log(\sqrt{a^2 + b^2}) + j \cdot tan^{-1}(\frac{b}{a})$ (C) Signup and view all the answers

According to De Moivre's theorem, if $z = r∠θ$, what is $z^n$?

$r^n∠nθ$ (B) Signup and view all the answers

What is the principal root of a complex number when finding its nth roots?

The root obtained when k = 0. (D) Signup and view all the answers

Which of the following correctly represents $sin(a + jb)$?

$sin(a)cosh(b) + jcos(a)sinh(b)$ (C) Signup and view all the answers

Which of the following correctly represents $arcsin(z)$?

$-j ln(jz + \sqrt{1 - z^2})$ (A) Signup and view all the answers

Which is the key characteristic of a 'zero matrix'?

All elements are zero. (C) Signup and view all the answers

What is the key property of a diagonal matrix?

The elements above and below the main diagonal are zeroes. (B) Signup and view all the answers

In an identity matrix, what are the values of the elements on the main diagonal?

Ones. (D) Signup and view all the answers

What condition defines a symmetric matrix A?

$A = A^T$ (C) Signup and view all the answers

What distinguishes a skew-symmetric matrix from a symmetric matrix?

Its transpose equals the negative of the matrix. (D) Signup and view all the answers

What is the defining property of an orthogonal matrix?

Its inverse equals its transpose. (B) Signup and view all the answers

What characterizes a Hermitian matrix?

It equals its conjugate transpose. (A) Signup and view all the answers

Which of the following is true of a skew-Hermitian matrix?

It is equal to the negative of its conjugate transpose. (D) Signup and view all the answers

What is the key property of a unitary matrix?

It is also called an orthogonal matrix. (C) Signup and view all the answers

An idempotent matrix A satisfies which condition?

$A^2 = A$ (B) Signup and view all the answers

What condition defines an involutory matrix A?

$A^2 = I$ (D) Signup and view all the answers

What is a periodic matrix and how do you define it?

Satisfies $A^k=A$, where k is the period (B) Signup and view all the answers

What are the conditions for matrix addition?

Matrices can only be added if their sizes are equal. (B) Signup and view all the answers

What are the conditions for matrix subtraction?

Matrices can only be subtracted if their sizes are equal. (A) Signup and view all the answers

For matrix multiplication, what must be equal?

The number of elements in a column of the first matrix should be equal to the number of elements in a row of the second matrix. (D) Signup and view all the answers

How can division of matrices be done given that there is no direct division?

Using inverse matrices. (B) Signup and view all the answers

What is a transpose of a matrix?

Where all the rows become column entries. (A) Signup and view all the answers

Which of the following statements is true about determinants?

If all elements of any row or column are zero, then determinant is equal to zero. (D) Signup and view all the answers

In the context of determinants, what does Cramer's Rule provide?

A formula for solving systems of linear equations using determinants. (C) Signup and view all the answers

Which of the following best describes a minor?

The determinant of a smaller matrix obtained by removing rows and columns from the original matrix . (C) Signup and view all the answers

What is the relationship between minors and cofactors?

Cofactors are minors multiplied by a phase factor (+1 or -1). (D) Signup and view all the answers

What is the key role of determinants in finding matrix inverses?

To evaluate the determinant of the matrix itself and dividing each of the elements in the transposed matrix. (A) Signup and view all the answers

What happens to an eigenvector when transformed by its matrix?

It scales. (B) Signup and view all the answers

Which equation defines how to calculate the eigenvalues?

$det(A-λI) = 0$ (C) Signup and view all the answers

Flashcards

Complex Numbers

Numbers extending real numbers with an imaginary component, expressed as a + bi.

j/i Operator

An imaginary unit defined as the square root of -1, used in complex numbers.