Problem 249 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Show that if \(A\) is an orthogonal matrix, then \(A^{T}\) is also orthogonal.

Step-by-Step Solution

Answer

The transpose of an orthogonal matrix is also orthogonal.

Step 1: Definition of Orthogonal Matrix

An \(n \times n\) matrix \(A\) is said to be orthogonal if \(A^{T}A=AA^{T}=I\), where \(A^{T}\) is the transpose of \(A\) and \(I\) is the identity matrix.

Step 2: Transpose of Orthogonal Matrix

If \(A\) is an orthogonal matrix, the transpose of \(A\), (\(A^{T}\)), and the inverse of \(A\) (let's call it \(A^{-1}\)), are equal. That is \(A^{T}=A^{-1}\). Therefore, \(A^{T}\) satisfies \(A^{T}A=AA^{T}=I\)

Step 3: Proof of Orthogonality of Transposed Matrix

Now, taking \(B\) as \(A^{T}\), we must show that \(B^{T}B=BB^{T}=I\). The transpose of \(B\), \(B^{T}\), is \(A\). Following the rules of transpose multiplication, \(B^{T}B\) equals \((A^{T})^T(A^{T})\), which simplifies to \(AA^{T}\). If \(A\) is orthogonal, this is the identity matrix \(I\). Similarly, \(BB^{T}\) equates to \(A^{T}A\), which is also \(I\). Therefore \(B^{T}B=BB^{T}=I\), so \(A^{T}\) is indeed orthogonal.

Key Concepts

Matrix Transpose

The transpose of a matrix is an operation that flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix. For a matrix A, its transpose is denoted as A^T. If A is an m x n matrix, then A^T will be an n x m matrix where the elements of the ith row of A become the elements of the ith column in A^T.

For example, if matrix A is:
\[ A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \]
Then the transpose, A^T, is:
\[ A^T = \begin{pmatrix} a & c \ b & d \end{pmatrix} \]

Understanding the transpose is crucial because it allows us to explore properties of matrices, such as symmetry and orthogonality. The transpose operation is central to the problem of proving that the transpose of an orthogonal matrix is also orthogonal.

Identity Matrix

An identity matrix, denoted as I, is a square matrix in which all the elements are zero except for the main diagonal ones, which are all one. This type of matrix acts as the multiplicative identity in the matrix world, akin to how the number '1' acts in regular multiplication. For matrix multiplication, any matrix A of appropriate size, when multiplied by the identity matrix, will result in the matrix A itself: AI = IA = A.

Here is an example of a 3x3 identity matrix:
\[ I = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} \]

The concept of the identity matrix is essential in understanding orthogonal matrices, as the defining property of an orthogonal matrix involves it equating to the identity matrix when multiplied by its transpose.

Proof of Matrix Properties

Proving matrix properties often involves demonstrating that certain conditions hold true under multiplication. When we talk about an orthogonal matrix A, we verify its orthogonality by showing that A^TA = AA^T = I. This means that multiplying the matrix by its transpose in either order yields the identity matrix, confirming its orthogonal nature.

The significance of this property is that the inverse of an orthogonal matrix is also its transpose, simplified as A^-1 = A^T. This elegant relationship makes computations more straightforward and is critical in many applications across geometry, physics, and computer graphics. Furthermore, by proving that this property holds for the transpose of an orthogonal matrix, we confirm that the set of orthogonal matrices is closed under the transpose operation.