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Inverses of Matrices

  • Understand the concept of matrix inverses and their geometric meaning.
  • Learn conditions for matrix invertibility using determinants.
  • Compute inverses of 2×2 and 3×3 matrices using formulas and row operations.
  • Apply matrix inverses to solve systems of linear equations.
  • Explore applications in data science and machine learning.

Definition of Matrix Inverse

The inverse of a square matrix \( A \) is a matrix \( A^{-1} \) such that:

\[ A A^{-1} = A^{-1} A = I \]

where \( I \) is the identity matrix. This means applying \( A \) followed by \( A^{-1} \) returns the original vector.

Conditions for Invertibility

A matrix is invertible if and only if its determinant is non-zero:

\[ \text{If } \det(A) \neq 0, \text{ then } A \text{ is invertible}. \]

Singular matrices (with zero determinant) do not have inverses.

Computing Matrix Inverses

For a 2×2 matrix:

\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]

For larger matrices, use Gaussian elimination or row-reduction to transform \( A \) into \( I \), while applying the same operations to \( I \) to obtain \( A^{-1} \).

Applications in Data Science

  • Solving linear systems: \( A\mathbf{x} = \mathbf{b} \Rightarrow \mathbf{x} = A^{-1}\mathbf{b} \)
  • Linear regression: computing coefficients using normal equations
  • Feature transformation and dimensionality reduction

Examples

Example 1: Find the inverse of:

\[ A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \]

Compute the determinant: \( \det(A) = 2 \cdot 4 - 3 \cdot 1 = 5 \)

\[ A^{-1} = \frac{1}{5} \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix} \]

Example 2: Use matrix inverse to solve:

\[ A\mathbf{x} = \mathbf{b}, \quad A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 5 \\ 6 \end{bmatrix} \]

Exercises

  • Question 1: Compute the inverse of \( A = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix} \).
  • Question 2: Determine if \( B = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix} \) is invertible.
  • Question 3: Solve \( A\mathbf{x} = \mathbf{b} \) using inverse, where \( A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix} \), \( \mathbf{b} = \begin{bmatrix} 4 \\ 5 \end{bmatrix} \).

  • Answer 1: \( A^{-1} = \frac{1}{(1)(5)-(2)(3)} \begin{bmatrix} 5 & -2 \\ -3 & 1 \end{bmatrix} = -1 \cdot \begin{bmatrix} 5 & -2 \\ -3 & 1 \end{bmatrix} \)
  • Answer 2: \( \det(B) = 2 \cdot 2 - 4 \cdot 1 = 0 \Rightarrow B \text{ is not invertible} \)
  • Answer 3: \( A^{-1} = \frac{1}{5} \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}, \quad \mathbf{x} = A^{-1} \mathbf{b} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \)

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