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Examples : Based on Properties of Determinants Video Lecture | Mathematics (Maths) Class 12 - JEE

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FAQs on Examples : Based on Properties of Determinants Video Lecture - Mathematics (Maths) Class 12 - JEE

1. What are the properties of determinants?
Ans. The properties of determinants include: - Multiplying a row or column by a scalar multiplies the determinant by the same scalar. - Swapping two rows or columns changes the sign of the determinant. - Adding a multiple of one row or column to another leaves the determinant unchanged. - If a matrix has a row or column of zeros, then its determinant is zero. - If two rows or columns of a matrix are equal, then its determinant is zero.
2. How can determinants be used to solve systems of linear equations?
Ans. Determinants can be used to solve systems of linear equations by using Cramer's Rule. Cramer's Rule states that the solution to a system of linear equations can be obtained by dividing the determinants of matrices formed from the coefficients of the variables. By evaluating the determinants and dividing them accordingly, the values of the variables can be found.
3. Can determinants be negative numbers?
Ans. Yes, determinants can be negative numbers. The sign of the determinant depends on the properties of the matrix. Swapping two rows or columns changes the sign of the determinant, so if a matrix has an odd number of row or column swaps, the determinant will be negative. If it has an even number of swaps, the determinant will be positive.
4. What happens if the determinant of a matrix is zero?
Ans. If the determinant of a matrix is zero, it means that the matrix is singular or non-invertible. This means that the matrix does not have a unique solution and the system of linear equations it represents may have either infinitely many solutions or no solution at all. This is an important condition to check when solving systems of equations using determinants.
5. Can determinants be used to find areas or volumes?
Ans. Yes, determinants can be used to find areas or volumes. In two-dimensional space, the absolute value of the determinant of a 2x2 matrix represents the area of the parallelogram formed by the column vectors of the matrix. In three-dimensional space, the absolute value of the determinant of a 3x3 matrix represents the volume of the parallelepiped formed by the column vectors of the matrix. By using determinants, geometric properties such as areas and volumes can be calculated.
204 videos|288 docs|139 tests

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