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FAQs on Determinants (Important Questions) Class 12th 2019

1. What is a determinant and why is it important in linear algebra?
Ans. A determinant is a scalar value that can be calculated from the elements of a square matrix. It is an important concept in linear algebra because it provides valuable information about the matrix. The determinant can determine whether a matrix has an inverse, whether a system of linear equations has a unique solution, and the volume or area scaling factor of a transformation.
2. How is the determinant of a matrix calculated?
Ans. The determinant of a matrix can be calculated using various methods, such as expansion by minors, cofactor expansion, or using a formula for 2x2 and 3x3 matrices. For larger matrices, the most efficient method is usually to use row operations to transform the matrix into an upper triangular form and then multiply the diagonal elements to find the determinant.
3. Can the determinant of a matrix be negative?
Ans. Yes, the determinant of a matrix can be negative. The sign of the determinant depends on the number of row or column swaps required to transform the matrix into an upper triangular form. If an even number of swaps is required, the determinant is positive; if an odd number of swaps is required, the determinant is negative.
4. What is the geometric interpretation of the determinant?
Ans. The determinant of a matrix has a geometric interpretation as the scaling factor of a linear transformation. For example, in two dimensions, the absolute value of the determinant represents the area scaling factor of a linear transformation, while the sign of the determinant determines whether the transformation preserves or reverses orientation. In three dimensions, the absolute value of the determinant represents the volume scaling factor, and the sign determines whether the transformation preserves or reverses orientation.
5. What are some real-life applications of determinants?
Ans. Determinants have various real-life applications. In physics, determinants are used in calculating moments of inertia, which determine how objects rotate around different axes. In computer graphics, determinants are used to determine whether a polygon is facing towards or away from the viewer, which is essential for rendering realistic images. Additionally, determinants are used in economics to solve systems of linear equations and analyze input-output models.
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