Properties of Transpose of the Matrices & Theorems related to Symmetric & Skew

Properties of Transpose of the Matrices & Theorems related to Symmetric & Skew Video Lecture | Mathematics (Maths) Class 12 - JEE

Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

FAQs on Properties of Transpose of the Matrices & Theorems related to Symmetric & Skew Video Lecture - Mathematics (Maths) Class 12 - JEE

 1. What are the properties of the transpose of a matrix?
Ans. The properties of the transpose of a matrix are as follows: 1. The transpose of the transpose of a matrix is the original matrix itself, i.e., (A^T)^T = A. 2. The transpose of the sum of two matrices is equal to the sum of their transposes, i.e., (A + B)^T = A^T + B^T. 3. The transpose of the product of two matrices is equal to the product of their transposes in reverse order, i.e., (AB)^T = B^T A^T. 4. The transpose of the scalar multiple of a matrix is equal to the scalar multiple of its transpose, i.e., (kA)^T = k(A^T), where k is a scalar. 5. The transpose of the inverse of a square matrix is equal to the inverse of its transpose, i.e., (A^(-1))^T = (A^T)^(-1), if A is invertible.
 2. What are the theorems related to symmetric matrices?
Ans. The theorems related to symmetric matrices are as follows: 1. Every square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. 2. The sum of two symmetric matrices is always a symmetric matrix. 3. The product of two symmetric matrices is not necessarily symmetric. 4. The transpose of a symmetric matrix is always symmetric. 5. The eigenvalues of a symmetric matrix are always real.
 3. What are the theorems related to skew-symmetric matrices?
Ans. The theorems related to skew-symmetric matrices are as follows: 1. The sum of two skew-symmetric matrices is always a skew-symmetric matrix. 2. The product of two skew-symmetric matrices is not necessarily skew-symmetric. 3. The transpose of a skew-symmetric matrix is also a skew-symmetric matrix. 4. The diagonal elements of a skew-symmetric matrix are always zero. 5. The eigenvalues of a skew-symmetric matrix are either zero or purely imaginary.
 4. How do transpose and symmetric matrices relate to each other?
Ans. The transpose of a symmetric matrix is always symmetric. If A is a symmetric matrix, then (A^T)^T = A, which means the transpose of the transpose is the original matrix itself. Therefore, the transpose of a symmetric matrix is symmetric.
 5. Can a matrix be both symmetric and skew-symmetric at the same time?
Ans. No, a matrix cannot be both symmetric and skew-symmetric at the same time. In a symmetric matrix, the diagonal elements must be the same as the corresponding elements across the diagonal. However, in a skew-symmetric matrix, the diagonal elements must be zero. These conditions are contradictory, so a matrix cannot satisfy both properties simultaneously.

Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

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