Determinants

Determinants

Edited By Team Careers360 | Updated on May 07, 2022 12:00 PM IST

Introduction:
Without understanding Determinants, learning about matrices is incomplete. The determinant of a matrix is determined by all of its components. The presence of a matrix's inverse is exactly proportional to the determinant's value. In Algebra, it is a highly helpful notion. A square matrix's determinant is a number that is exclusively specified for square matrices. Determinants are mathematical objects that may be used to solve and analyse systems of linear equations. Determinants are used in a variety of fields including engineering, science, economics, and social science. They may be used to solve linear equations, see how a linear transformation changes volume or area, and modify variables in integrals. Furthermore, we may think of determinants as a function with a square matrix as its input and an integer as its output. So far, we've learned what determinants are, how they're expressed, and how they're used. Let's have a look at the Properties of Determinants, which will assist us in simplifying our assessment by determining the maximum amount of zeros in a row or column. These features apply to any order of determinants. We will, however, limit ourselves to determinants of order 3.

Latest: JEE Main: high scoring chapters | Past 10 year's papers

Don't Miss: Most scoring concepts for NEET | NEET papers with solutions

New: Aakash iACST Scholarship Test. Up to 90% Scholarship. Register Now

List of topics according to NCERT and JEE Main/NEET syllabus:

Determinant of a matrix
Properties of determinants
Minor of determinants
Cofactor of determinant
Area of triangle using determinants
Adjoint of matrix
Adjoint of matrix
Inverse of matrix
Solution of linear equation

Important concepts and Laws:

We may correlate a value (real or complex) called the determinant of the square matrix A with any square matrix A = [aij] of order n, where a = (i, j)th member of A. This may be thought of as a function that assigns a unique number to each square matrix (real or complex). If M is a collection of square matrices, K is a set of real or complex integers, and f: M → K is defined by f (A) = k, where A ∈ M and k ∈ K, then f (A) is known as the determinant of A. It is also denoted by | A | or det A or Δ. If A = [a] is a one-dimensional matrix, then A's determinant is equal to a.

Properties of determinants

Property 1: If the rows and columns are swapped, the determinant's value stays intact.

Property 2: If the signs of any two rows (or columns) of a determinant are swapped, the determinant's sign changes.

Property 3: The value of a determinant is zero if any two rows (or columns) of the determinant are equal (all related elements are the same).

Property 4: If each element of a determinant's row (or column) is multiplied by a constant k, the value of the determinant is multiplied by k.

Property 5: A determinant can be expressed as the sum of two (or more) determinants if some or all of the elements in a row or column are stated as the sum of two (or more) determinants.

Property 6: If the equivalents of similar elements from other rows (or columns) are added to every element of a determinant's row or column, the determinant's value remains the same.

A cofactor is a number obtained by subtracting the column and row of a certain element from a matrix.

A minor is the determinant of a square matrix that is formed by eliminating one column and one row from a bigger square matrix.

A matrix's adjoint is also known as the matrix's adjugate. It refers to the transpose of that specific matrix's cofactor matrix. The denotation of adjoint for a matrix A is adj (A). The inverse of a matrix A, on the other hand, is a matrix that produces an identity matrix when multiplied by matrix A.

The entire space encompassed by a particular triangle can be defined as the area of a triangle. The area of a triangle is calculated by multiplying the base by the height. After that, double it by two. Because a parallelogram may be divided into two triangles, the division by two is used. Multiplication of the lengths of the two sides is required to get the area of any right triangle. These two sides are parallel to one another. After that, half of it must be taken.

NCERT Notes Subject Wise Link:

Importance of determinants in class 11

The NCERT answer is a critical component of the CBSE Curriculum as well as the demands of students. In a simple style, these Solutions cover the whole course as well as all of the fundamentals. In prior years' examinations, students might discover duplicate questions from NCERT exercises. As a result, using these solutions to answer problems should be a significant component of test preparation. There are 82 problems in Chapter 4 of class 12 Maths that cover all of the subjects connected to determinants. NCERT Solutions is a collection of step-by-step answers to all of the questions in the textbook. As a result, students find these Solutions to be quite helpful in their test preparations. Chapter 4 is about determinants, and it's a simple chapter if pupils get the concept. These are the most effective instruments for kids to grasp the fundamentals. This chapter explains how to determine physical dimensions such as area and volume using determinants. As a result, pupils should choose NCERT Solutions.

NCERT Solutions Subject wise link:

NCERT Exemplar Solutions Subject wise link:

Frequently Asked Question (FAQs)

1. Is it possible for a determinant to be negative?

A determinant is a real integer that is not a matrix in most cases. A determinant, on the other hand, can be a negative number. Most notably, it has no resemblance to absolute value other than the usage of vertical lines.

2. What happens if the determinant is set to 0?

If the determinant of a square matrix n n A is 0, then A is not invertible in general. Furthermore, if a matrix's determinant is not zero, the linear system it represents is linearly independent. When a matrix's determinant is 0, the rows become linearly dependent vectors, and the columns become linearly dependent vectors.

3. What use does Cramer's rule serve?

It's a method for solving systems of equations with the same number of equations as variables that use determinants. It also takes into account a two-variable system of two linear equations.

4. Are positive matrix determinants always positive?

A positive definite matrix's determinant must always be positive. As a result, a non-singular matrix is always a positive definite matrix. The inverse of a positive definite matrix is also a positive definite matrix.