Application of Determinants and Matrices: Overview, Questions, Preparation

Updated on Aug 19, 2025 12:18 IST

The last topic under the chapter determinants is the applications of matrices and determinants. The concepts of adjoint and inverse of a matrice has numerous advantages and offers practical uses not just in the field of mathematics but in our daily lives as well. Researchers and scientists in the innovation of technology that have benefitted the humankind as a whole have used the concepts learned through this chapter. Over the years, various new technologies have emerged, which combine the principles of physics and mathematics for advance development of new devices and strategies for solving complex equations. Let us dive into these real life uses through this article.

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Real Life Uses of Determinants and Matrices

Real Life Uses of Determinants and Matrices

Here are some of the widely recognized uses of matrices and determinants that have been used by the humans in revolutionizing today’s world:

Linear and Differential Equations:

Matrices can be easily used to solve linear equations of the form:

AX = B

Where,

A = coefficient matrice

B = constant matrice

X = unknown matrice

Differential equations can be solved using inverse matrices by arranging the points in a matrice form. Adjoints of a matrice are also used to compute inverse of a matrice using determinants through the formula:

A−1= Adj(A)/det(A)

Computer Graphics:

In 2D/3D graphics and animation, properties of matrices are used to perform -

Scaling
Rotation
Translation
Reflection

These properties are highly useful in designing AR/VR technologies, video games, IoT sensors, etc.

Cryptography:

Also known as Hill Cipher, this is a technique where encryption and decryption techniques require matrices for their functioning. Adjoints and inverse matrices are used in such scenarios to ensure security at the user end.

Electrical Engineering:

In the field of physics, formulas of electrical circuits use matrices for defining variables and solve the complex equations to calculate voltage. For example: Kirchhoff’s Law.

Markov Chains:

In probability, inverse and transition matrices are used to define changes in the state and compute steady states as well as the states which change over the time (markov processes).

Business:

Economic models and systems are created with the help of matrices and determinants, which help understand the supply chain, demand in the market and steps that can be taken to maximize profit as per the requirement of the market.

Coordinate Geometry:

Inverse matrices are also used in geometry to find equations of transformation and points on the graph that help compute the points of the line.

Cramer’s Rule:

This is one of the most important uses of determinants. Systems of linear equations can be solved with the help of determinants using the Cramer’s rule, which states that:

xi= det(Ai)/ det(A)

Where,

Ai = obtained by replacing ith column of the matrice A.

Area and Volume:

This is an important chapter of the subject mathematics from which questions are frequently asked in competitive exams like the JEE MAINS. Determinants can also be used to calculate area of parallelograms, volume of parallelepiped, jacobian determinants in calculus, 3D vectors, etc.

Eigenvalues and Stability Analysis:

The characteristic equation is used to compute eigenvalues and eigenvectors, which is used in geometry, physics, calculate stability, electrical circuits and mechanics, control systems etc. This equation can be represented as:

det(A−λI)=0