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Chapter Analysis
Intermediate26 pages • EnglishQuick Summary
The chapter on determinants in Class 12 Mathematics deals with the definition, properties, and applications of determinants up to order three for matrices with real entries. It explains how to calculate determinants using expansion along rows or columns, and introduces concepts like minors, cofactors, and adjugate matrices. The chapter further applies determinants to solving systems of linear equations and finding the area of triangles.
Key Topics
- •Definition of Determinant
- •Properties of Determinants
- •Minors and Cofactors
- •Application in Solving Linear Equations
- •Area of a Triangle using Determinants
- •Cramer's Rule
- •Adjugate and Inverse of Matrices
- •Consistency of System of Equations
Learning Objectives
- ✓Understand the calculation of determinants for matrices up to order three
- ✓Learn the significance and properties of determinants
- ✓Calculate minors and cofactors
- ✓Solve linear equations using determinants
- ✓Compute the area of triangles using determinants
Questions in Chapter
Evaluate the determinants in Exercises 1 and 2.
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Choose the correct answer in Exercise 17 to 19.
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Find area of the triangle with vertices at the point given in each of the following: (i) (1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8) (iii) (–2, –3), (3, 2), (–1, –8)
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Additional Practice Questions
What is the cofactor of an element in a determinant?
mediumAnswer: The cofactor of an element a_ij is denoted by A_ij and defined as A_ij = (–1)^(i+j) M_ij, where M_ij is the minor of the element a_ij.
How do you determine the area of a triangle using determinants?
mediumAnswer: The area of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3) can be calculated using the formula: 1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) |. This is equivalent to the absolute value of a determinant constructed with the coordinates.
Explain the method of expansion of determinants along a row.
mediumAnswer: To expand a determinant along a row, multiply each element of the row by its corresponding cofactor and sum the results. This method can be applied to any row or column.
What does the determinant of a matrix signify geometrically?
hardAnswer: Geometrically, the determinant of a matrix can represent the scaling factor of the linear transformation described by the matrix. For a 2x2 matrix, it represents the area of the parallelogram spanned by its column vectors.
How can determinants be used to determine if a set of points are collinear?
easyAnswer: If three points are collinear, the area of the triangle formed by them is zero, which implies that the determinant constructed from their coordinates is zero.