Final answer:
The transpose of an orthogonal matrix is its inverse because the definition of an orthogonal matrix involves its transpose multiplying the matrix to yield the identity matrix. The same applies in reverse, satisfying the conditions for being an inverse.
Step-by-step explanation:
The question pertains to why the transpose of an orthogonal matrix is its inverse. An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (orthonormal vectors). This means that when the matrix is multiplied by its transpose, the result is the identity matrix.
To understand why the transpose of an orthogonal matrix is its inverse, consider the definition of an orthogonal matrix Q. The defining property of an orthogonal matrix is that Q multiplied by its transpose QT equals the identity matrix I, i.e., QQT = I. This equation also implies QTQ = I, where QT is the transpose of Q.
In the realm of matrices, the inverse of a matrix A is another matrix, denoted as A-1, such that when A is multiplied by A-1, the result is the identity matrix, AA-1 = I. Therefore, if multiplying an orthogonal matrix Q by its transpose QT yields the identity matrix, it can be concluded that QT serves as the inverse of Q, because it satisfies the condition for an inverse matrix.
To illustrate this with an example, consider the orthogonal matrix Q:
- If we take the transpose of Q, we get QT.
- Multiplying Q by QT we get the identity matrix, confirming that Q is an orthogonal matrix: QQT = I.
- Since the result is the identity matrix, we deduce that the transpose QT acts as the inverse of Q.
Therefore, for an orthogonal matrix, the concept of a transpose and an inverse are the same, which is a distinctive property of orthogonal matrices in linear algebra.