NCERT Solutions for Class 12 Maths Chapter 4
Class XII Maths Chapter 4 solutions is given here. Download here the NCERT Solutions for class 12 Maths chapter 4 Determinants all exercises with miscellaneous exercise in Hindi Medium and English Medium in PDF for the new academic session 2020-2021. UP board solutions for class 12 Maths Chapter 4 are same as the NCERT Sols for 12th Maths Chapter 4. UP Board students are also using following NCERT Textbooks for new session, so class 12 UP Board students also can use these solutions for their help in problem solving. Other boards like MP Board, Uttarakhand, Gujrat Board, etc., are also following the textbooks based on NCERT.The the solutions of questions are helpful for all the students who are using NCERT Books. To understand the 12th Maths Chapter 4 properly, we must know everything about matrices. So, first learn the concepts of the chapter 3 Matrices to do Chapter 4 Determinants.
NCERT Solutions for class 12 Maths chapter 4
|Class: 12||Maths (English and Hindi Medium)|
12th Maths Chapter 4 Solutions
In this chapter we shall study determinants up to order three only with real entries. The history about determinant is given below to know more about this fact. Download NCERT solutions for class 12 Maths chapter 4 determinants all six exercise with miscellaneous in PDF format. UP Board Solutions for Class 12 Maths Chapter 4 is same as NCERT Sols for 12th Maths Chapter 4 Determinants. So, UP Board students can also take the benefits of these solutions.
Class 12 Maths Chapter 4 Solutions in Hindi Medium
Class 12 Maths Chapter 4 Solutions in PDF
- Download Class 12 Maths Exercise 4.1 in PDF
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- Download Class 12 Maths Exercise 4.3 in PDF
- Download Class 12 Maths Exercise 4.4 in PDF
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- Download Class 12 Maths Exercise 4.6 in PDF
- Download Class 12 Maths Miscellaneous Exercise 4
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Class 12 Maths Exercise 4.1 Videos Solutions
Class 12 Maths Exercise 4.2 Videos Solutions
Class 12 Maths Exercise 4.3 Videos Solutions
Class 12 Maths Exercise 4.4 Videos Solutions
Class 12 Maths Exercise 4.5 Videos Solutions
Class 12 Maths Exercise 4.6 Videos Solutions
Class 12 Maths Miscellaneous Exercise 4 Videos Solutions
Properties of Determinants
If all the rows of a determinant are converted into the corresponding columns, the value of the determinant remains same.
If two rows (columns) of a determinant are interchanged, the value of the new determinant is the additive inverse of the value of the given determinant.
The value of a determinant gets multiplied by k, if every entry in any of its row (column) is multiplied by k.
If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be express as sum of two (or more) determinants.
If the corresponding entries in any two rows ( or columns) are identical, the value of the determinant is zero.
The value of a determinant does not changes if any of its rows (columns) is multiplied by non-zero real number k and added to another row (column).
MINORS AND COFACTORS
Minor – Removing entries of the column and the row containing a given element of a determinant and keeping the surviving entries as they are, yields a determinant called the minor of the given element.
Cofactor – If we multiply the minor of an element by (-1)^(i+j), where i is the number of the row and j is the number of the column containing the element, then we get the cofactor of that element.
1. The Chinese early developed the idea of subtracting columns and rows as in simplification of a determinant using rods. Seki Kowa, the greatest of the Japanese Mathematicians of seventeenth century in his work ‘Kai Fukudai no Ho’ in 1683 showed that he had the idea of determinants as well as their expansion.
2. T. Hayashi, “The Fakudoi and Determinants in Japanese Mathematics,” in the proc. of the Tokyo Math. Soc., V. Vendermonde was the first to recognise determinants as independent functions. He may be called the formal founder.
3. Laplace (1772), gave general method of expanding a determinant in terms of its complementary minors.
4. Lagrange, in 1773, treated determinants of the second and third orders and used them for purpose other than the solution of equations.
5. Gauss, in 1801, used determinants in his theory of numbers.
6. Jacques – Philippe – Marie Binet, in 1812, stated the theorem relating to the product of two matrices of m-columns and n-rows, which for the special case of m = n reduces to the multiplication theorem.
7. Cauchy, in 1812, presented one on the same subject. He used the word ‘determinant’ in its present sense. He gave the proof of multiplication theorem more satisfactory than Binet’s.
8. The greatest contributor to the theory was Carl Gustav Jacob Jacobi, after this the word determinant received its final acceptance.
Important Questions on 12th Maths Chapter 4
For any square matrix A, |A| satisfies the following properties.
(i) |A′| = |A|, where A′ = transpose of matrix A.
(ii) If we interchange any two rows (or columns), then sign of the determinant changes.
(iii) If any two rows or any two columns in a determinant are identical (or proportional), then the value of the determinant is zero.
(iv) Multiplying a determinant by k means multiplying the elements of only one row (or one column) by k.
(v) If we multiply each element of a row (or a column) of a determinant by constant k, then value of the determinant is multiplied by k.
(vi) If elements of a row (or a column) in a determinant can be expressed as the sum of two or more elements, then the given determinant can be expressed as the sum of two or more determinants.
(vii) If to each element of a row (or a column) of a determinant the equimultiples of corresponding elements of other rows (columns) are added, then value of determinant remains same.