Applications of Determinants and Matrices and Solved Examples

Ans: The above-given system of equations can be depicted in the form, AX = B, 

Where,

A = [1 3 2 6]

X = [x y]

B = [5 8]

Checking the determinant of A,

\(\)\(\mid{A}\mid\) = 6-6=0g

We must check (adj A) B, for checking the consistency of the system

adj A= [6 -3 -2 1]

(adj A)B= [6 -3 -2 1 5 8]

(adj A) B = [30- 24 -10+ 8]

(adj A) B = [6 -2]

The solution to the given system of linear equations does not exist since (adj A) B ≠ 0. As a result, the system of equations is inconsistent.

Ans: AX = B can be written as a solution to the above system of equations.

A= [2 1 2 3]

X= [x y]

B= [3 6]

Finding \(\mid{A}\mid\), \(\mid{A}\mid\) = 40

So, the system of equations is consistent, possesses a unique solution and can be found using X = A-1B

A^-1= \(\frac{adj A}{\mid{A}\mid}\)

A^-1= ¼ [3 -1 -2 2]

X= A^-1B

[x y] =1/4 [3 -1 -2 2][ 3 6]

[x y] =1/4 [9 -6 -6+ 12]

[x y] =1/4 [3 6]

From this, we can get to a conclusion that,

x=3/4 

And y=6/4

Ans: The above given system of equations can be written in the form AX=B, where

A= [1 -1 2 3 4 -5 2 -1 3]

X= [x y z]

B= [7 -5 12]

Checking the determinant of A

|A| = 1 * (12 – 5) + 1 * (9 + 10) + 2 * (-3 – 8)

|A| = 7 + 19 – 22 = 4

|A| ≠ 0

As a result, its inverse exists, and so X =A^-1B can be used to find a unique answer

A^-1= \(\frac{adj A}{\mid{A}\mid}\)

A^-1= ¼ [7 1 -3 -19 -1 11 -11 -1 7]

X =A^-1B

[x y z] =¼ [7 1 -3 -19 -1 11 -11 -1 7][ 7 -5 12]

[x y z] =¼ [49- 5- 36 -133+ 5+ 132 -77+ 5+ 84]

[x y z] = ¼ [8 4 12]

[x y z] = [2 1 3]

You can see that x = 2, y = 1, and z = 3 

As a result, x = 2, y = 1, and z = 3 are the values.

Ans: We know that |A|(t) = e^t= cos²t+= sin²t)= e^t≠0, hence A(t)^-1 exists

Cofactor matrix is given by

C(t) = [1 0 0 0 e^tcos ^t e^tsin^t 0 – e^tsin^t e^tcos^t]

Hence the inverse is 

1/e^t = [1 0 0 0 e^tcos^t e^tsin^t 0 – e^tsin^t e^tcos^t ]^T

= [e^t 0 0 0 cos^t – sin^t 0 sin^t cos^t]

Ans: Given, 

5x – y + 4z = 5

2x + 3y + 5z = 2

5x – 2y + 6z = –1

The above given system of equations can be written in the form AX=B, where

[1 -1 2 3 4 -5 2 -1 3 x y z] = [5 2 -1]

When we compare,

A= [5 -1 4 2 3 5 5 -2 6]

The determinant of A is given by,

\(\mid\)A\(\mid\) = \(\mid\)5 -1 4 2 3 5 5 -2 6\(\mid\)

\(\mid\)A\(\mid\) = 5 (18+10) + 1(12-25) + 4(-4-15)

=140-13-76

=140-89

=51

Therefore\(\mid\)A\(\mid\)≠0

So, the system of equations is consistent.

Ans: If the matrix equations AX=b and |A|≠0 is equal, the matrix is square, and the inverse exists.

A^-1 = 1/|A| [A11 A21 An1 \(\vdots { } \vdots { } \ddots { } \vdots\) A1n A2n \(\cdots\)Ann]

and

X = A^-1B

Unless, of course, we're talking about a field. If the equation is in integers, a system with a positive determinant and no solution is inevitable. For instance, if A= (2) and b= (1)

Ques. When (P³) =P, why is the determinant of matrix P equal to zero? (5 marks)

Ans: In general, assuming that P's entries are in a field, it is not true that if P³ =P , then P's determinant is zero.

If P³ = P, 

then 

P³ - P=0, P(P² – I) = 0

or 

P(P + I)(P - I)= 0

According to the Cayley-Hamilton theorem, P's minimum polynomial must divide x(x-1) (x+1) (or be equal to it). 

This suggests that the smallest polynomial might be (x1) (x+1), or (x-1) or (x+1).

P² = I, 

Hence P is an involution, a matrix whose inverse is itself if the smallest polynomial is (x-1) (x+1) (for example, it represents a reflection or a rotation by half a revolution). 

If x-1 is the smallest polynomial, then P = I. 

If x+1 is the minimum polynomial, then P= -I. 

All of these are possible examples of matrices with a nonzero determinant and satisfy P³ = P.

Ans: Neither. A determinant isn't a norm, and a norm isn't a determinant.

Norms must be non-negative, which the determinant is not, and they must meet the triangle inequality, which the determinant fails to accomplish.

Ans: A fundamental concept in graph theory is the adjacency matrix of a finite graph. It keeps track of which edges connect the graph's vertices. Logical matrices are matrices that contain only two separate values (1 and 0 indicating "yes" and "no," respectively). The distance (or cost) matrix records the distances between the edges.

These notions can be used to web pages linked by hyperlinks, cities interconnected by highways, and so on, where the matrices tend to be sparse, containing few nonzero entries (unless the connection network is extraordinarily dense). As a result, network theory can benefit from especially specialized matrix algorithms.

Graph Theory

Ans: A stochastic matrix is a square matrix that is used to explain the transitions of a Markov chain in mathematics. Each of its elements is a probability represented by a non-negative real number. It is also known as a Markov matrix, a probability matrix, a transition matrix, or a substitution matrix.

Stochastic Matrix

Andrey Markov invented the stochastic matrix in the early twentieth century, and it has since been used in a wide range of scientific subjects, including probability theory, statistics, mathematical finance, and linear algebra, as well as computer science and population genetics. There are several distinct types and definitions of stochastic matrices:

• A right stochastic matrix is a real square matrix whose rows add up to one.
• A left stochastic matrix is a real square matrix with 1 as the sum of each column.
• A square matrix of nonnegative real numbers with each row and column summing to 1 is known as a doubly stochastic matrix.

Ans: The description of linearly connected harmonic systems is a common use of matrices in physics. The kinetic term is given by multiplying a generalized velocity by a mass matrix, while the interactions are defined by multiplying a force matrix by a displacement vector.

The best technique to find solutions is to diagonalize the matrix equation and find the system's eigenvectors, or normal modes. When it comes to the internal dynamics of molecules, or the internal vibrations of systems made up of mutually bound component atoms, techniques like this are critical. They're also required for characterizing mechanical vibrations and electrical circuit oscillations.

More matrix applications can be found in geometrical optics. The wave aspect of light is ignored in this approximative theory. As a result, light beams are indeed geometrical rays in this paradigm. If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as ray transfer matrix analysis, which involves multiplying a two-component vector with a two-by-two matrix: the vector's components are the light ray's slope and distance from the optical axis, while the matrix encodes the optical element's properties.

A refraction matrix describes refraction at a lens surface, and a translation matrix describes the translation of the plane of reference to the next refracting surface, where another refraction matrix applies. The matrix resulting from the product of the components' matrices simply describes the optical system, which consists of a combination of lenses and/or reflective elements.