📘 Chapter 2: Inverse and Rank of a Matrix

🔁 Inverse of a Matrix

• The inverse of a matrix \( A \) exists only if \( A \) is square and nonsingular.
• If \( A^{-1} \) exists: \[ A \cdot A^{-1} = A^{-1} \cdot A = I \]
• Methods to compute:
  • Adjoint: \( A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A) \)
  • Gauss–Jordan elimination

Example

\[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \Rightarrow \det(A) = -2 \]
\[ \text{adj}(A) = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \Rightarrow A^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \]

📏 Rank of a Matrix

• Rank is the maximum number of linearly independent rows/columns.
• \( \text{rank}(A) \) is the dimension of the row or column space.
• It helps identify:
  • Solvability of systems \( Ax = b \)
  • Whether the matrix is invertible

Example

\[ B = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \Rightarrow \text{Row 2} = 2 \cdot \text{Row 1} \Rightarrow \text{rank}(B) = 1 \]

🧠 Key Takeaways

• If \( \det(A) = 0 \), then \( A^{-1} \) doesn’t exist
• Full rank ⇒ invertible ⇒ non-zero determinant
• Rank tells us how much information a matrix encodes

📝 Practice Problem

Q: Find the inverse of \[ A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \]

Solution:

• \( \det(A) = 5 \)
• \( A^{-1} = \frac{1}{5} \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix} \)