Determinant Of The Identity Matrix Proof

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Sep 22, 2024

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The determinant of the identity matrix is a fundamental concept in linear algebra, and its proof is essential for understanding the properties of determinants. The identity matrix, denoted by I, is a square matrix with ones on the main diagonal and zeros elsewhere. Its determinant, denoted as |I|, plays a crucial role in various mathematical operations and is always equal to 1. This article will delve into the proof of the determinant of the identity matrix, exploring its significance and the steps involved.

Understanding the Determinant

The determinant of a square matrix is a scalar value that provides valuable information about the matrix, including its invertibility, singularity, and the volume scaling factor in linear transformations. It can be calculated using various methods, including cofactor expansion, row reduction, and the Leibniz formula. For the identity matrix, its determinant can be proven through a direct calculation using the cofactor expansion method.

Proof of the Determinant of the Identity Matrix

Let's consider an n x n identity matrix I. The proof involves demonstrating that |I| = 1 by using cofactor expansion along the first row:

| I | = 1 * C₁₁ + 0 * C₁₂ + 0 * C₁₃ + ... + 0 * C_1n

Where:

• C_ij represents the cofactor of the element in the i-th row and j-th column.
• C₁₁ is the cofactor of the element in the first row and first column.

Since all elements except the first element in the first row are zeros, all cofactors except C₁₁ become zero. Therefore, the determinant simplifies to:

| I | = C₁₁

Now, let's calculate C₁₁:

C₁₁ = (-1)¹⁺¹ * |M₁₁|

Where:

• M₁₁ is the minor of the element in the first row and first column, which is the (n-1) x (n-1) matrix obtained by removing the first row and first column from I.

Since M₁₁ is also an identity matrix of size (n-1) x (n-1), its determinant, |M₁₁|, can be calculated using the same cofactor expansion method recursively. This process will eventually lead to a 1x1 identity matrix, whose determinant is simply 1.

Therefore:

C₁₁ = (-1)² * 1 = 1

And finally:

| I | = C₁₁ = 1

Significance of the Result

The determinant of the identity matrix being equal to 1 has several significant implications:

• Invertibility: A matrix is invertible if and only if its determinant is non-zero. Since the determinant of the identity matrix is 1, the identity matrix is always invertible.
• Scaling Factor: In linear transformations, the determinant of a matrix represents the scaling factor of the volume of the transformed space. The identity matrix represents no transformation, so its determinant of 1 indicates no scaling.
• Matrix Operations: The identity matrix plays a crucial role in matrix operations like multiplication and inversion. The fact that its determinant is 1 ensures that these operations do not change the fundamental properties of the matrices involved.

Conclusion

The proof of the determinant of the identity matrix being equal to 1 is a foundational result in linear algebra. It highlights the unique properties of the identity matrix and its importance in various mathematical operations. Understanding this proof is crucial for comprehending the concepts of determinants, matrix invertibility, and linear transformations. The determinant of the identity matrix serves as a fundamental building block for more advanced mathematical concepts and applications in various fields, such as physics, engineering, and computer science.