Problem 25 - Algebra for JEE ADVANCED

Question

If \(A\) is an orthogonal matrix, then \(A^{-1}\) equals a. \(A^{T}\) b. \(A\) c. \(A^{2}\) d. none of these

Step-by-Step Solution

Answer

The answer is (a) \( A^{T} \).

Step 1: Understanding Orthogonal Matrices

A matrix is orthogonal if its columns are orthonormal vectors. A fundamental property of orthogonal matrices is that the transpose of the matrix is equal to its inverse. We need to establish this property mathematically.

Step 2: Defining Orthogonal Matrix Property

If matrix \( A \) is orthogonal, then \( A^T A = I \), where \( I \) is the identity matrix. This relationship implies that the inverse of \( A \) must satisfy \( A A^{-1} = I \).

Step 3: Applying Transpose Property

Given the property \( A^T A = I \), we can see that multiplying \( A \) from the right gives \( A^T A = A^T \), which implies \( A^{-1} = A^T \).

Step 4: Choosing the Correct Option

Since we have established that \( A^{-1} = A^T \), option (a) \( A^{T} \) is the correct answer. It aligns with the property that the inverse of an orthogonal matrix is its transpose.

Key Concepts

Matrix Transpose

The transpose of a matrix is an important concept in linear algebra. When you transpose a matrix, you essentially "flip" it over its diagonal. This means the rows of the original matrix become the columns of the transposed matrix, and vice versa. If a matrix is denoted by \( A \), its transpose is written as \( A^T \).

Here’s a simple example:

• For a matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \), the transpose would be \( A^T = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix} \).

This operation is particularly useful when dealing with orthogonal matrices. An interesting property of orthogonal matrices is that the transpose is also the inverse, which we'll discuss further. Understanding the transpose is crucial for comprehending how matrices interact and transform.

Inverse Matrix

A matrix's inverse can be thought of as the "undoing" of a matrix operation. If you have a matrix \( A \), its inverse is denoted as \( A^{-1} \). The relationship between a matrix and its inverse is such that when you multiply them together, you get the identity matrix \( I \). Mathematically, this is expressed as \( A A^{-1} = A^{-1} A = I \).

Not every matrix has an inverse. A matrix must be square (same number of rows and columns) and have a non-zero determinant for its inverse to exist. Orthogonal matrices are a special case where the inverse has a unique property. For orthogonal matrices, the inverse is the same as the transpose. This simplifies many calculations because you can substitute the transpose straight away.

Ultimately, understanding inverses is key to solving many matrix equations. They allow you to reverse or manipulate matrix transformations in mathematical and physical applications.

Orthonormal Vectors

Orthonormal vectors are the building blocks of orthogonal matrices. When we say vectors are orthonormal, we mean they have two specific properties: they are orthogonal to each other, and each vector has a magnitude of 1.

Orthogonal means that the vectors are at right angles (90 degrees) to each other. If \( \mathbf{u} \) and \( \mathbf{v} \) are two vectors, they are orthogonal if their dot product is zero: \( \mathbf{u} \cdot \mathbf{v} = 0 \). Normalization to a magnitude of 1 is achieved by dividing each vector by its magnitude.

• Example: If \( \mathbf{u} = \begin{bmatrix} 3 \ 4 \end{bmatrix} \), its magnitude is 5. The normalized vector is \( \mathbf{u} = \begin{bmatrix} \frac{3}{5} \ \frac{4}{5} \end{bmatrix} \).

Orthonormal vectors form the columns of an orthogonal matrix, giving it its defining properties. They ensure the matrix transforms spaces in ways that preserve distance and angles.

Identity Matrix

The identity matrix is like the number 1 in matrix operations. It is a square matrix with ones on the main diagonal and zeros elsewhere. For any matrix \( A \), when you multiply it by an identity matrix \( I \), the result is the original matrix. This is expressed as \( AI = IA = A \).

In the 2x2 case, the identity matrix looks like this:

• \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \)

The identity matrix is crucial for defining the inverse. Remember, for an inverse matrix \( A^{-1} \), the equation \( A A^{-1} = I \) always holds true. This gives the identity matrix a vital role in calculations involving inverses and orthogonal matrices.

Overall, the identity matrix is a key concept in understanding matrix algebra and how transformations interact with each other.