# Higher Algebra Hall & Knight Determinants (Chapter 33) Solutions

Hall and Knight Higher Algebra Solutions for Chapter 33 ‘Determinants’ have been created in order to help you gain complete conceptual clarity in solving the questions of this chapter. The topics covered in this chapter are the eliminant of two and three homogeneous linear equations. You will learn to interchange the rows and columns in a matrix and the development of determinant of the third order, the change in sign when two rows or columns in a determinant is are interchanged, and the result when two rows or columns are identical and other results related to operations on determinants. You will learn to find the product of two determinants besides learning how to reduce a determinant, different cases of constituents, determinants of fourth-order, and any other.

There are 2 exercises in this chapter comprising 46 questions in total. Solving these exercise questions will help you build a robust understanding of matrices and determinants. These topics carry heavy weightage in your Class 12 board exams along with competitive exams. We have provided easy explanations in our solutions for Higher Algebra By H.S. Hall and S.R. Knight for a complete understanding of the chapter. These questions range from the beginner to the advanced level. You can use these exercise questions for practising and revising topics suitable for competitive exams like IIT JEE and NEET.

The subject matter experts at Instasolv has created this set of solutions for Higher Algebra by Hall & Knight with great care using the simple and understandable language for explaining the concepts of Determinants of the third, fourth, and n^{th} order. You will find assistance at every step with precise reasoning. The solutions of this classic algebra book have been written in a format strictly compliant to the latest competitive exam pattern and guidelines.

## Important Topics Covered under Hall and Knight Higher Algebra Solutions Chapter 33: Determinants

**Introduction**

We will discuss briefly the determinants as well as the elementary properties of determinants in this chapter.

Let us assume, two homogeneous linear equations

a_{1 }x + b_{1 }y = 0 and a_{2 }x + b_{2 }y = 0, then by multiplying the first equation with b_{2}, second with b1 and subtracting and dividing by x, we get

a_{1 }b_{2 }–a_{2 }b_{1 }= 0. This is also written in the following notation,

Such an expression is called a determinant. Since its expanded form consists of two terms, it is known as the determinant of second order. a_{1}, b_{1}**, **b_{2}, and **a _{2}** are constituents and a

_{1}b

_{2}, a

_{2}b

_{1}are the elements.

**Elemental Properties of Determinants**

- If we interchange the rows and columns of a given determinant, then there will be no change in the value of the determinants.
- There will be no change in the value of the determinant, while the sign of the determinant will change after we interchange two rows or two columns in the given determinant.
- If there are two rows or two columns in a given determinant, which are exactly the same, the value of the determinant becomes null.
- If we multiply the determinant by a factor, then each element of any row or any column will have to be multiplied by the same factor.
- If we multiply two determinants, the product will be a determinant.

**The determinant of the Third Order**

The determinant of the third-order can be expressed as,

For the homogenous linear equations, a_{1}x +b_{1}y + c_{1}z = 0 , a_{2}x + b_{2}y + c_{2}z = 0, and a_{23}x + b_{3}y + c_{3}z = 0

The elements can be mentioned as,

a_{1}b_{2}c_{3 }– b_{3}c_{2 }+ b_{1}a_{3}c_{2 }– c_{2}a_{2 }+ c_{1}a_{2}b_{3 }– a_{3}b_{2}

We must note that after we interchange all the rows into columns and all the columns into rows, then there is no change in the value of the determinant.

### Exercise Discussion of Hall and Knight Higher Algebra Solutions Chapter 33: Determinants

- In the exercise questions of this chapter, you will get to solve both formulae based as well as concept-based questions.
- Exercise 33.a comprises 28 questions of different nature such as proofs of important results and identities, evaluation of equations, word problems, and questions based on finding the values of determinants.
- In exercise 33.b, the questions are such that you will be required to find the values of the given determinants besides proving some important results. You will also get to solve a word problem to show that the determinant of nth order with certain conditions evaluates to be equal to unity.

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