Problem 48 - Single Variable Calculus

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a matrix, then \(\left(A^{T}\right)^{T}=A\).

Step-by-Step Solution

Answer

The statement is true. If \(A\) is a matrix, then \(\left(A^{T}\right)^{T}=A\). This is because reversing the indices twice through transposition cancels out the changes and brings the matrix back to its original configuration.

Step 1: Understanding Matrix Transposition

Transposing a matrix means interchanging its rows and columns. Mathematically, if a matrix \(A\) has elements \(a_{ij}\) and is of size \(m \times n\), then its transpose \(A^T\) will have elements \(a_{ji}\) and be of size \(n \times m\).

Step 2: Analyzing \(\left(A^{T}\right)^{T}\)

To find the transpose of the transpose of \(A\), we first need to find \(A^T\). Once we have the matrix \(A^T\), we will then find its transpose, i.e., \(\left(A^{T}\right)^{T}\).

Step 3: Comparing \(\left(A^{T}\right)^{T}\) and \(A\)

We will now compare the elements of \(\left(A^{T}\right)^{T}\) and \(A\) to see if they are equal. If the elements of both matrices match, then the statement is true, and we can explain why it is true. Otherwise, the statement is false, and we must provide an example demonstrating its falsehood. The element at position \(i,j\) in \(\left(A^{T}\right)^{T}\) can be denoted by \(a^{T^{T}}_{ij}\). According to our definition of matrix transposition, we first transpose A to get elements \(a^{T}_{ij}=a_{ji}\), and then take the transposition of this \(A^{T}\) matrix: \(a^{T^{T}}_{ij}=a^T_{ji}\). But \(a^T_{ji}=a_{ij}\), since reversing the index twice essentially brings us back to the original position. Therefore, \(a^{T^{T}}_{ij}=a_{ij}\), and the elements of \(\left(A^{T}\right)^{T}\) and \(A\) match for all positions.

Step 4: Conclusion

Since the elements of \(\left(A^{T}\right)^{T}\) are equal to the elements of A, we can conclude that the given statement is true: If \(A\) is a matrix, then \(\left(A^{T}\right)^{T}=A\). This is because reversing the indices twice through transposition cancels out the changes and brings the matrix back to its original configuration.