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7-6. Cofactor Expansion (Linear Algebra for Data Science)

Previous:    • 7-5. Properties of Determinants  (Lin...   Next:    • 8-1. Systems of Linear Equations (Lin...   Playlist:    • Linear Algebra for Data Science   In this chapter of our linear algebra journey, we dive into the world of determinants—specifically, the powerful technique of cofactor expansion, also known as Laplace expansion. Whether you're tackling a 3x3 matrix or an n x n matrix, calculating the determinant can seem daunting. But with a systematic method like cofactor expansion, the process becomes both manageable and logical. The story begins with the question: How do we calculate the determinant of a square matrix using only one row or column? The answer lies in breaking down a larger matrix into smaller parts—by expanding along a single row or column, we reduce an n x n matrix to smaller (n−1) x (n−1) matrices. This recursive approach lies at the heart of cofactor expansion. To make the abstract formula more tangible, we walk through a concrete example using a 3x3 matrix. We start by expanding the first row, then compute the minors, which are the 2x2 determinants formed by removing the relevant row and column. Next, we apply the sign alternation pattern, a checkerboard of positive and negative signs, ensuring each term contributes correctly. A common question arises: Do we have to expand across every row? Fortunately, no. The determinant is invariant, meaning the final value remains the same regardless of the row or column chosen for expansion. This allows us to choose the most convenient row—ideally, one with the most zeros—to simplify computation. The video also unpacks the sigma notation, explaining how it helps sum the contributions of each element in the chosen row or column, each paired with its cofactor (the minor times the sign). This methodical approach leads to an accurate and efficient calculation of the determinant. Whether you're a student preparing for exams, someone brushing up on matrix algebra, or just exploring the foundations of linear transformations and vector spaces, this tutorial will guide you step-by-step through the logic and technique behind cofactor expansion. 【Linear Algebra for Data Science】 This course explains the knowledge of linear algebra, which is essential for data science enthusiasts, in an easy-to-understand way. In this channel, you can obtain: 1. Intuitive understanding 2. Mathematical understanding 3. Practical understanding of linear algebra for data science! Many linear algebra concepts are abstract, and it is challenging to understand how they are applied in real-world data analytics. Lectures in this channel provide easy-to-understand explanations of various concepts of linear algebra in a way that is not only mathematically but also intuitively understandable. In addition, the lectures explain how each concept is useful in data analytics as a practical application. Instructor: Takuma Kimura (木村 琢磨), Ph.D. Scientist of Organizational Behavior and Business Analytics https://orcid.org/0000-0001-7126-188X   / takuma-kimura-ba6242104   #linearalgebra #datascience #cofactorexpansion #laplaceexpansion #machinelearning