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Properties of Determinants of Matrices Video Lecture | Mathematics for Competitive Exams
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FAQs on Properties of Determinants of Matrices Video Lecture - Mathematics for Competitive Exams
| 1. What is the determinant of a matrix? |  |
Ans. The determinant of a matrix is a scalar value that can be calculated for square matrices only. It represents certain properties of the matrix and is denoted by "det(A)" or "|A|".
| 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 row operations. The specific method used depends on the size of the matrix and the available information.
| 3. What are the properties of determinants of matrices? | |
Ans. The properties of determinants of matrices include:
1. Property of Linearity: The determinant of a sum of matrices is equal to the sum of their determinants.
2. Property of Scalar Multiplication: Multiplying a matrix by a scalar multiplies its determinant by the same scalar.
3. Property of Transposition: The determinant of a matrix is the same as the determinant of its transpose.
4. Property of Row or Column Interchange: Interchanging any two rows or columns of a matrix changes the sign of its determinant.
5. Property of Row or Column Multiplication: Multiplying any row or column of a matrix by a scalar multiplies its determinant by the same scalar.
| 4. How does the determinant of a matrix relate to invertibility? | |
Ans. The determinant of a matrix is directly related to its invertibility. If the determinant is non-zero, the matrix is said to be invertible or non-singular. In this case, the matrix has a unique inverse. On the other hand, if the determinant is zero, the matrix is said to be singular or non-invertible, and it does not have an inverse.
| 5. 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 interchanges required to transform the matrix into its row-echelon form. If an odd number of interchanges is performed, the determinant is negative; otherwise, it is positive.
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Nov 22, 2024 Last updated |
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