If a matrix is symmetric, which of the following must be true? (2021)
Practice Questions
Q1
If a matrix is symmetric, which of the following must be true? (2021)
A = A^T
A = -A^T
A^2 = I
A^T = 0
Questions & Step-by-Step Solutions
If a matrix is symmetric, which of the following must be true? (2021)
Step 1: Understand what a matrix is. A matrix is a rectangular array of numbers arranged in rows and columns.
Step 2: Learn about the transpose of a matrix. The transpose of a matrix A, denoted as A^T, is formed by flipping A over its diagonal. This means that the row and column indices of each element are swapped.
Step 3: Define a symmetric matrix. A matrix A is called symmetric if it is equal to its transpose, which means A = A^T.
Step 4: Recognize that for a matrix to be symmetric, every element in the matrix must satisfy the condition A[i][j] = A[j][i] for all i and j, where A[i][j] is the element in the ith row and jth column.
Step 5: Conclude that if a matrix is symmetric, it must always satisfy the condition A = A^T.
Symmetric Matrix – A matrix A is symmetric if it is equal to its transpose, i.e., A = A^T.
