A=[ 1+z^2 1 1 0 1+z^2 1 0 0 1+z^2 ] and B=[ 1+z^-2

A=[ 1+z^2 1 1 0 1+z^2 1 0 ...

Key Concepts

Matrix Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether the matrix is invertible (nonzero determinant) and the scaling factor of the linear transformation associated with the matrix. In many problems, setting the determinant to zero is used to find critical values or parameters that change the behavior of the linear system.

Triangular Matrices

A triangular matrix is a type of matrix that has all zero entries either above or below the main diagonal. The determinant of a triangular matrix is simply the product of its diagonal elements, which greatly simplifies computations. This property is often exploited when analyzing matrices that inherently or through manipulation have triangular form.

Complex Numbers on the Unit Circle

Complex numbers that lie on the unit circle satisfy the condition that their magnitude is one. This unique property implies that such numbers are of the form z = e^(i?), where ? represents the argument of z. The unit circle property leads to useful relationships, such as the fact that the multiplicative inverse of z is its complex conjugate, which simplifies analysis in problems involving complex expressions.

Argument of a Complex Number

The argument of a complex number is the angle, typically measured in radians, between the positive real axis and the line representing the complex number in the complex plane. It is a fundamental concept in complex analysis and is critical when expressing complex numbers in polar form. Determining the argument is central to solving problems where the phase or orientation of a complex number affects the outcome.

Transcript

00:01 In the given question we are told that we have two matrices that are given as the first matrix is given as 1 plus z square 1 1 1 0 1 plus z square 1 0 0 1 plus z square this is the first matrix that is given and the second matrix that is given as b which is given as 1 plus z raised to minus 2 0 0 1 plus z raised to minus 2 0 and 1 1 plus z raised to minus 2 and 1 1 1 plus z raised to minus 2 so these are the two matrices that are given in the christian and we are told that z is a complex number with the modulus of z equal to 1 and we are told we are asked if the modulus of or the determinant of a plus b if the determinant of a plus b is equal to 0 what is the argument of the complex number z so this is what we asked what we are asked right so what we do over here is we can take both these matrices and add them right so once we add them what we would get is the determinant of a plus b would be equal to the determinant of 2 plus z square plus z raise 2 minus 2 can be written as 1 by z square and 1 1 1 1 2 plus z square 1 and 1 1 2 plus z square plus 1 by z square so this is what we get as the determinant of a plus b b right so this is 2 plus z square plus 1 by z square is simply the expansion of the term of the expression x plus 1 by z whole square right 1 1 1 z whole square right 1 1 z plus 1 by z whole square 1 and the third row we have 1 1 z plus 1 by z whole squared so this is what we have as the simplified result and we should know that we can write z as let's say z can be written as r times cos theta plus i sine theta right where theta is the argument of the complex number z and r is the modulus right so now what we can do is if z is equal to r cost theta plus one i.

03:21 X x is simply the conjugate or we can write r cost theta minus i sine theta which means when we take z plus 1 by z is simply the conjugate or we can write r cost theta minus i sine theta which means when we take z plus 1 by z what we have is 2 r cost theta right so if we have this and r is the modulus of z modulus of the complex number z we can write this is equal to 2 cost theta right so let's take this and substitute it in the above matrix so 2 cost theta whole square would be equal to 4 cosquare theta 1 1 4 cos square theta 4 cosquare theta 1 and 1 1 4 cause square 2 theta so this is what we have as the determinant and in the next step what we are going to do is we can take we can take this we can do a column operation which is c1 changes to c1 plus c2 plus c3 and this will give us 2 plus 4 cosquare theta 2 plus 4 cosquare theta 2 plus 4 cosquare theta in the first column and the rest is unchanged so now we can take 2 plus cos square theta 4 cos square theta as a common factor so we will have 2 plus 4 4 co square theta as a common factor so we will have 2 plus 4 square theta times one one one one one we have over here four cause square theta one one four cause square theta so this is what we have now in the next step what we do is we can expand this determinant and what we will get is two plus four cost square theta times 4 cost square theta 4 cost square theta minus 1 the whole square square so this is equal to 0 right so now what we are going to do is we can equate 2 plus 4 cos square theta equal to 0 from which we can write cos square theta is equal to 1 by 4 and we can also write from this what we can write is cost theta is equal to plus or minus 1 by 2 right so the value or the argument the angle of the angle theta for which cost theta is plus or minus 1 by 2 it is not 1 by root 2 it is 1 by 2 plus or minus 1 by root 2 plus or 2 is the angle 2 pi by 3, right? so since we have in the question we have an option which says the argument of z is equal to 2 by 3, 2 pi by 3 which is given by option c 2 pi by 3, we can write the argument of z is in fact equal to 2 pi by 3...

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