Problem 46 - A Complete Resource Book in Mathematics for JEE Main

Question

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

Step-by-Step Solution

Answer

A) 0

Step 1: Understanding the Condition

The given condition is that the matrix \( M \) is an orthogonal matrix because \( M^{\prime} M = I \). Additionally, \( \det(M) = 1 \). This means \( M \) is a special orthogonal matrix, often denoted as having determinant +1.

Step 2: Calculate \( \det(M-I) \)

To find \( \det(M-I) \), we need to understand that since \( M \) is an orthogonal matrix with \( \det(M) = 1 \), the eigenvalues of \( M \) are on the unit circle. Since \( \det(M) = 1 \), the eigenvalues can be \( 1, e^{i\theta}, e^{-i\theta} \) where \( \theta eq 0 \) because eigenvalue 1 should appear only once to satisfy \( M - I \).

Step 3: Determine Eigenvalues Contribution to \( \det(M-I) \)

The determinant \( \det(M-I) \) is the product of the eigenvalues of \( M-I \). The eigenvalues of \( M-I \) will be \( 0, e^{i\theta} - 1, e^{-i\theta} - 1 \). Thus, the determinant becomes \( (0)(e^{i\theta} - 1)(e^{-i\theta} - 1) = 0 \).

Step 4: Conclude with the Correct Choice

The value of \( \det(M-I) \) is 0, which corresponds to option (A).

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that pops out from a square matrix. Think of it like a unique fingerprint for matrices that helps in solving systems of linear equations, among other things. It tells us a lot about the matrix.

• If the determinant is zero, the matrix does not have an inverse.
• It can give us insight into the volume of transformation described by the matrix.
• For a matrix representing a linear transformation, its determinant tells if the transformation preserves orientation and volume.

For a matrix \( M \), to find the determinant of \( M - I \) where \( I \) is the identity matrix, think of adjusting the matrix by reducing diagonal elements by 1, then compute the determinant. This determinant is closely linked to the eigenvalues of the matrix. Calculating determinants is crucial in understanding the behavior of matrices and the systems they represent.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are concepts that help us understand matrices better. They are like a magnifying glass, showing the internal movements within a matrix operation.
In simple terms:

• An eigenvector of a matrix \( M \) is a non-zero vector that only changes by a scalar factor when the matrix is applied to it.
• The scalar factor is known as the eigenvalue.

When you have a matrix like \( M \) that is orthogonal with \( \det(M)=1 \,\) its eigenvalues lie on the unit circle. This means that all calculations of how vector directions change are rotation transformations.
For an orthogonal matrix, these eigenvalues can be specific values such as \( 1, e^{i\theta}, \text{and} e^{-i\theta} \). This deeply influences the determinant of matrices like \( M-I \), since it’s about the eigenvalue differences.

Special Orthogonal Matrices

Special orthogonal matrices hold a special place in the matrix world. When a matrix \( M \) is orthogonal and its determinant is 1, it's truly geometric. These matrices describe rotations and reflections in space without distorting objects.

• These matrices preserve lengths and angles.
• Their determinant is always +1, distinguishing them from generic orthogonal matrices which can also flip directions.
• They are closely related to rotations in space, especially in 3D, making them crucial for understanding physical phenomena.

One way to verify if a matrix belongs to this category is checking if \( M^\prime M = I \) and \( \det(M) = 1 \). Such matrices help maintain the rigidity of objects when transformed, which is an essential concept in graphics and physics.