Problem 7 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(M\) is a \(3 \times 3\) matrix, where \(M^{\prime} M=I\) and \(\operatorname{det}(M)=1\), then \(\operatorname{det}(M-I)=\) (A) 1 (B) 0 (C) \(-1\) (D) None of these

Step-by-Step Solution

Answer

Option (B) 0.

Step 1: Interpret the Given Information

The problem states that the matrix \(M\) satisfies the condition \(M^{\prime}M = I\), where \(I\) is the identity matrix. This implies that \(M\) is an orthogonal matrix.

Step 2: Recognize Important Characteristics of an Orthogonal Matrix

For an orthogonal matrix \(M\), it is known that \(M^{\prime} = M^{-1}\). Additionally, the determinant of an orthogonal matrix satisfies \(|\operatorname{det}(M)|=1\). Since we are given \(\operatorname{det}(M) = 1\), \(M\) is a special orthogonal matrix.

Step 3: Calculate \(\operatorname{det}(M-I)\)

To find \(\operatorname{det}(M-I)\), we use the fact that the eigenvalues of an orthogonal matrix with determinant \(1\) are \(1, \cos\theta + i\sin\theta, \cos\theta - i\sin\theta\) (complex conjugates) or some combination leading to real determinant possibilities in this context. Since we have 1 as an eigenvalue, \(M - I\) will have determinant zero because the matrix becomes singular due to this eigenvalue.

Step 4: Draw the Conclusion Using Algebra and Properties

Since an eigenvalue of 1 in \(M-I\) makes it singular, the determinant is 0. Hence, after factoring and analyzing the characteristic polynomial, \(\operatorname{det}(M-I) = 0\).

Key Concepts

Orthogonal Matrix

An orthogonal matrix is a special type of square matrix that holds a unique property. This property is that when an orthogonal matrix is multiplied by its own transpose, the result is the identity matrix, or simply put,

• If \( A \) is an orthogonal matrix, then \( A^\prime A = I \), where \( I \) is the identity matrix.

In simpler terms, this means that the rows (and columns) of an orthogonal matrix are orthogonal unit vectors. Because these vectors are orthogonal and of unit length, the transformations that orthogonal matrices represent do not affect the length of the vectors they act upon. They are typically used in computations needing efficient computations, like computer graphics and simulations, because they conserve angles and lengths. These matrices also possess the property that their inverse is equal to their transpose, that is,

• \( A^{-1} = A^\prime \)

Eigenvalues

Eigenvalues are particular numbers associated with a matrix, crucial in understanding the full picture of what a matrix represents. An eigenvalue is essentially a scalar that illustrates how a matrix can be transformed without changing its original direction when applied to an eigenvector. For orthogonal matrices, especially one with a determinant of 1, they have very distinctive properties:

• The eigenvalues preserve norm and include complex conjugates.
• The eigenvalue representation, typically, could include a real eigenvalue, say 1, and a pair of complex conjugates \( \cos\theta + i\sin\theta \) and \( \cos\theta - i\sin\theta \).

This specific scenario occurs because orthogonal matrices are reflections and rotations in nature. When a scalar transformation \( M - I \) is introduced, obtaining eigenvalue(s) which equal zero means the matrix \( M-I \) is singular. This ultimately leads to \( \operatorname{det}(M-I) = 0 \). Eigenvalues offer insights into the matrix structure, revealing if it can stretch or squash dimensions in specific directions.

Special Orthogonal Matrix

A special orthogonal matrix is a refined version of an orthogonal matrix with an additional important property. This type of matrix not only satisfies the condition of being orthogonal, but its determinant is specifically equal to 1. The determinant condition leads to a subtly significant implication in physical applications, hinting at a transformation that involves only rotations without reflections. Hence:

• An orthogonal matrix is special when \( \operatorname{det}(M) = 1 \).

This minor yet crucial distinction makes special orthogonal matrices relevant in fields that require understanding of rotation in 3D spaces, like physics and engineering. For a matrix \( M \), being special orthogonal translates to creating transformations that do not invert the handedness of space. Moreover, the fact that these matrices have determinants of 1 implies that they preserve volume, making them integral in transformations conserving scale and orientation.