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# NCERT Solutions for Class 12 Maths chapter 4 Determinants

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 free download in pdf format. This chapter consists of various properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle, adjoint and inverse of a square matrix, consistency and inconsistency of system of linear equations and solution of linear equations in two or three variables using inverse of a matrix. All the contents related to determinants like assignments, notes, tests, previous year questions, etc. will be uploaded in this page following the  syllabus 2017-18. Back to NCERT Solutions

## NCERT Solutions for class 12 Maths chapter 4 Determinants

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.

### Mixed Chapter Tests

Chapter 1, 2, 3 & 4

Level 1  Test 1

Level 2  Test 1

Level 3  Test 1    Test 2

#### 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). 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

• 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.
• 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.
• Laplace (1772), gave general method of expanding a determinant in terms of its complementary minors.
• Lagrange, in 1773, treated determinants of the second and third orders and used them for purpose other than the solution of equations.
• Gauss, in 1801, used determinants in his theory of numbers.
• 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.
• 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.
• The greatest contributor to the theory was Carl Gustav Jacob Jacobi, after this the word determinant received its final acceptance.