NCERT Solutions for Class 12 Maths Chapter 4

NCERT Solutions for class 12 Maths chapter 4 Determinants exercise 4.1, 4.2, 4.3, 4.4, 4.5, 4.6 and miscellaneous exercises in Hindi Medium and English Medium download in PDF for new academic session 2020-21.

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NCERT Solutions for class 12 Maths chapter 4

Class:12
Subject:Maths
Chapter 4:Determinants

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 exercise 4.1, 4.2, 4.3, 4.4, 4.5, 4.6 and miscellaneous exercises in PDF form given below.









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.



Historical Facts

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

Which type of matrices have determinants?
Only square matrices have determinants.
What are the properties of Determinants?
Properties of Determinants
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.
What will be the value of a determinant, if all the elements of a row (or column) are zeros?
If all the elements of a row (or column) are zeros, then the value of the determinant is zero.