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Transpose of a Matrix - Matrices Video Lecture | Mathematics (Maths) Class 12 - JEE
FAQs on Transpose of a Matrix - Matrices Video Lecture - Mathematics (Maths) Class 12 - JEE
| 1. What is the transpose of a matrix? |  |
Ans. The transpose of a matrix is a new matrix formed by interchanging its rows with columns. In other words, if the original matrix has dimensions m x n, the transpose will have dimensions n x m.
| 2. How do you find the transpose of a matrix? | |
Ans. To find the transpose of a matrix, you simply need to interchange its rows with columns. For each element in the original matrix, its corresponding element in the transpose will be at the position given by the column index of the original element as the row index and vice versa.
| 3. What are the properties of the transpose of a matrix? | |
Ans. The transpose of a matrix has the following properties:
- The transpose of the transpose of a matrix is the original matrix.
- The transpose of a sum of matrices is the sum of their transposes.
- The transpose of a product of matrices is the product of their transposes in reverse order.
| 4. Can the transpose of a matrix be the same as the original matrix? | |
Ans. Yes, a matrix can be its own transpose if it is a symmetric matrix. A matrix is symmetric if it is equal to its transpose, i.e., A = A^T. Symmetric matrices have interesting properties and are widely used in various mathematical applications.
| 5. How does the transpose of a matrix affect its determinants and eigenvalues? | |
Ans. The transpose of a matrix does not change its determinants. In other words, if the original matrix has a determinant value of x, the transpose will also have a determinant of x. However, the eigenvalues of a matrix and its transpose are the same. The eigenvalues represent the scaling factors of the eigenvectors, and they remain unchanged when the matrix is transposed.
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