The Main Idea

• Matrix Inverses:
  • For a square matrix [latex]A[/latex], its inverse [latex]A^{-1}[/latex] satisfies: [latex]AA^{-1} = A^{-1}A = I[/latex]
  • [latex]I[/latex] is the identity matrix, with [latex]1[/latex]s on the main diagonal and [latex]0[/latex]s elsewhere
  • Not all square matrices have inverses; those that do are called invertible
• Properties of Matrix Inverses:
  • Only square matrices can have inverses
  • If [latex]A[/latex] is invertible, its inverse [latex]A^{-1}[/latex] is unique
  • The identity matrix acts like [latex]1[/latex] in matrix algebra: [latex]AI = IA = A[/latex]
• Identity Matrix:
  • 2×2 Identity Matrix: [latex]I_2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}[/latex]
  • 3×3 Identity Matrix: [latex]I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}[/latex]
• Solving Systems of Equations:
  • For a system [latex]AX = B[/latex], the solution is [latex]X = A^{-1}B[/latex]
  • This method is efficient for larger systems of equations
• Verifying Inverses:
  • To show that [latex]B[/latex] is the inverse of [latex]A[/latex], prove that [latex]AB = BA = I[/latex]

The Main Idea

• Definition of Matrix Inverse:
  • For a square matrix [latex]A[/latex], its inverse [latex]A^{-1}[/latex] satisfies: [latex]AA^{-1} = A^{-1}A = I[/latex]
  • [latex]I[/latex] is the identity matrix
• Methods to Find Matrix Inverses:
  • Matrix Multiplication Method
  • Augmenting with Identity Method
  • Inverse Formula (for 2×2 matrices)
• Matrix Multiplication Method:
  • Set up equation: [latex]A \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}[/latex]
  • Solve system of equations to find [latex]a, b, c, d[/latex]
• Augmenting with Identity Method:
  • Create augmented matrix [latex][A|I][/latex]
  • Perform row operations to transform [latex]A[/latex] into [latex]I[/latex]
  • The transformed I becomes [latex]A^{-1}[/latex]
• Inverse Formula for 2×2 Matrices:
  • For [latex]A = \begin{bmatrix} a & b \ c & d \end{bmatrix}[/latex]
  • [latex]A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}[/latex]
  • Only valid if [latex]ad-bc \neq 0[/latex]

The Main Idea

• Definition of Matrix Inverse:
  • For a square matrix [latex]A[/latex], its inverse [latex]A^{-1}[/latex] satisfies: [latex]AA^{-1} = A^{-1}A = I[/latex]
  • [latex]I[/latex] is the 3×3 identity matrix: [latex]I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}[/latex]
• Method for Finding 3×3 Matrix Inverse:
  • Augment the original matrix with the identity matrix
  • Perform row operations to transform the left side into the identity matrix
  • The right side becomes the inverse matrix
• Steps to Find 3×3 Matrix Inverse:
  • Write the augmented matrix [latex][A|I][/latex]
  • Use elementary row operations to transform [latex]A[/latex] into [latex]I[/latex]
  • The resulting right side is [latex]A^{-1}[/latex]
• Singularity:
  • Not all 3×3 matrices have inverses
  • A matrix with no inverse is called singular
  • If row operations result in a row of zeros, the matrix is singular
• Verification:
  • Check that [latex]AA^{-1} = A^{-1}A = I[/latex]

The Main Idea

• Matrix Representation of Systems:
  • [latex]AX = B[/latex], where:
    • [latex]A[/latex]: Coefficient matrix
    • [latex]X[/latex]: Variable matrix
    • [latex]B[/latex]: Constant matrix
• Solution Method:
  • Multiply both sides by [latex]A^{-1}[/latex]: [latex]A^{-1}AX = A^{-1}B[/latex]
  • Simplify: [latex]IX = A^{-1}B[/latex]
  • Solution: [latex]X = A^{-1}B[/latex]
• Matrix Inverse Properties:
  • [latex]A^{-1}A = AA^{-1} = I[/latex]
  • Not all matrices have inverses (singular matrices)
• Order Matters:
  • [latex]A^{-1}B \neq BA^{-1}[/latex] (matrix multiplication is not commutative)