Problem 76 - Algebra for JEE ADVANCED

Question

Which of the following is an orthogonal matrix? a. \(\left[\begin{array}{ccc}6 / 7 & 2 / 7 & -3 / 7 \\ 2 / 7 & 3 / 7 & 6 / 7 \\\ 3 / 7 & -6 / 7 & 2 / 7\end{array}\right]\) b. \(\left[\begin{array}{ccc}6 / 7 & 2 / 7 & 3 / 7 \\ 2 / 7 & -3 / 7 & 6 / 7 \\\ 3 / 7 & 6 / 7 & -2 / 7\end{array}\right]\) c. \(\left[\begin{array}{ccc}-6 / 7 & -2 / 7 & -3 / 7 \\ 2 / 7 & 3 / 7 & 6 / 7 \\\ -3 / 7 & 6 / 7 & 2 / 7\end{array}\right]\) d. \(\left[\begin{array}{ccc}6 / 7 & -2 / 7 & 3 / 7 \\ 2 / 7 & 2 / 7 & -3 / 7 \\\ -6 / 7 & 2 / 7 & 3 / 7\end{array}\right]\)

Step-by-Step Solution

Answer

Matrix (a) is the orthogonal matrix.

Step 1: Understand Orthogonal Matrix

A matrix is orthogonal if its transpose is equal to its inverse. This means that for any orthogonal matrix \( A \), \( A^T A = I \) where \( A^T \) is the transpose of \( A \) and \( I \) is the identity matrix.

Step 2: Transpose and Multiply for Matrix a

Calculate the transpose \( A^T \) of matrix (a), and then find \( A^T A \). If the result is the identity matrix, then this is an orthogonal matrix. For matrix (a): \[A = \begin{bmatrix} \frac{6}{7} & \frac{2}{7} & -\frac{3}{7} \ \frac{2}{7} & \frac{3}{7} & \frac{6}{7} \ \frac{3}{7} & -\frac{6}{7} & \frac{2}{7} \end{bmatrix}\]\( A^T A = I \), confirm if it holds.

Step 3: Transpose and Multiply for Remaining Matrices

Repeat the process for matrices b, c, and d. Calculate the transposes for each matrix, \( B^T, C^T, \) and \( D^T \). Then multiply each by its original matrix (e.g., \( B^T B \)) to check if the product equals the identity matrix for each. None of these will equal the identity matrix.

Step 4: Conclusion on Orthogonal Matrix

Matrix (a) satisfies the condition \( A^T A = I \), confirming it is orthogonal, whereas the others do not. Therefore, matrix (a) is the orthogonal matrix.

Key Concepts

Transpose of a Matrix

The transpose of a matrix is a fundamental concept in linear algebra. It involves swapping the rows and columns of a matrix. For a given matrix \( A \), its transpose is denoted as \( A^T \). You create \( A^T \) by flipping all the elements of \( A \) over its diagonal.
For example, if \( A \) is a \( 2 \times 3 \) matrix, with elements:

• Row 1: [a, b, c]
• Row 2: [d, e, f]

Then \( A^T \) would be a \( 3 \times 2 \) matrix:

• Row 1: [a, d]
• Row 2: [b, e]
• Row 3: [c, f]

Understanding the transpose is crucial when working with orthogonal matrices, as the property \( A^T A = I \) is central to verifying if a matrix is orthogonal.

Matrix Multiplication

Matrix multiplication involves taking two matrices and producing a new matrix. It is not the same as multiplying individual numbers; the process depends heavily on the rows of the first matrix and the columns of the second.
Consider two matrices \( A \) and \( B \) where \( A \) is of size \( m \times n \) and \( B \) is of size \( n \times p \). The resulting matrix \( C = AB \) will be of size \( m \times p \).
The element \( c_{ij} \) of matrix \( C \) is computed as follows:

• Multiply each element of the \( i \)-th row of \( A \) with the corresponding element of the \( j \)-th column of \( B \), then take the sum of these products.

Matrix multiplication is a pivotal part of solving exercises related to orthogonal matrices because you multiply the matrix with its transpose to check if it yields the identity matrix.

Inverse of a Matrix

The inverse of a matrix \( A \) is a matrix that, when multiplied with \( A \), results in the identity matrix. Not all matrices have inverses; a matrix must be square (same number of rows and columns) and have a non-zero determinant to possess an inverse.
If \( A^{-1} \) is the inverse of \( A \), then both \( A A^{-1} = I \) and \( A^{-1} A = I \) must hold, where \( I \) is the identity matrix.
For diagonal, triangular, or certain structured matrices, finding an inverse might be easier, but for arbitrary matrices, algorithms like Gauss-Jordan elimination or matrix adjugates are typically used.
In the context of orthogonal matrices, \( A^T \) serves as the inverse of \( A \), showcasing a unique property exclusive to orthogonal matrices.

Identity Matrix

The identity matrix \( I \) is a special form of a square matrix. It has 1s along its main diagonal and 0s elsewhere. For a \( n \times n \) identity matrix, it looks like this:

• Row 1: [1, 0, 0, ..., 0]
• Row 2: [0, 1, 0, ..., 0]
• ...
• Row n: [0, 0, 0, ..., 1]

The identity matrix acts like the number 1 in matrix operations. When any matrix is multiplied by its corresponding identity matrix, it remains unchanged.
Therefore, \( AI = A \) and \( IA = A \) for any matrix \( A \).
In the case of orthogonal matrices, checking whether the multiplication of a matrix and its transpose equals the identity matrix confirms its orthogonality.

Properties of Orthogonal Matrices

Orthogonal matrices have fascinating properties that make them useful in various mathematical and applied fields. One key property is that an orthogonal matrix preserves the length of vectors and the angle between them during transformations.
Some remarkable attributes include:

• For an orthogonal matrix \( A \), it holds that \( A^T = A^{-1} \). This means the transpose of the matrix is the same as its inverse.
• When you multiply an orthogonal matrix by its transpose, you get an identity matrix: \( A^T A = I \).
• The determinant of an orthogonal matrix is always \( +1 \) or \( -1 \).
• Orthogonal matrices are square and their column vectors are orthonormal, meaning they are perpendicular and unit vectors.

These properties make orthogonal matrices crucial in processes like rotating coordinate systems and preserving data during transformations.