Notes on Matrix Multiplication and Inverses

Matrix Multiplication

Overview

• Matrix multiplication has different methods that yield the same result.
• Important to understand how to multiply matrices and how to find inverses.

Basic Rules for Matrix Multiplication

• To multiply two matrices A and B, producing a result C:
  • Each entry Cᵢⱼ is found by taking the dot product of row i of matrix A and column j of matrix B.
  • Formula for the entry C₃₄: C₃₄ = ∑(A₃ₖ * Bₖ₄) for k from 1 to n.

Conditions for Multiplication

• For matrices A (m x n) and B (n x p):
  • Number of columns in A (n) must equal the number of rows in B (n).
  • Resulting matrix C will have dimensions m x p.

Alternative Methods of Matrix Multiplication

1. Column Method:

   • Each column of B is multiplied by A to get the respective columns of C.
   • Resulting columns are linear combinations of the columns of A.

2. Row Method:

   • Each row of A is multiplied by the entire rows of B to yield each row of C.
   • Resulting rows are linear combinations of the rows of B.

3. Column Times Row Method:

   • Multiply a column from A by a row from B, resulting in a full-size matrix.
   • Example: Multiply column [2,3,4] with row [1,6] to yield a matrix with multiples of respective vectors.

4. Block Multiplication:

   • Allows breaking down matrices into smaller block matrices for multiplication.
   • If A and B are partitioned into blocks A1, A2, B1, B2, then calculate C block by block.

Inverses of Matrices

Basic Definition

• An inverse of a matrix A (if it exists) is denoted by A⁻¹.
• Condition for existence: A * A⁻¹ = I (identity matrix).
• If A has a left inverse, it is also a right inverse for square matrices.

Conditions for Invertibility

1. Non-singular vs. Singular:

   • A matrix is called non-singular (invertible) if it has an inverse.
   • A matrix is singular if it does not have an inverse.

2. Examples of Non-invertible Matrices:

   • For instance, the matrix

     | 1 3 |
     | 2 6 |
     

     has no inverse because its determinant is 0 and the columns are linear combinations.

3. Column Combinations:

   • If some combination of columns results in the zero column, the matrix is non-invertible.

Finding the Inverse of a Matrix

1. Example:

   • Consider matrix A:

     | 1 3 |
     | 2 7 |
     

   • Use Gauss-Jordan elimination to find the inverse.

2. Gauss-Jordan Elimination:

   • Augment matrix A with the identity matrix and perform row operations.
   • Continue until the left side forms an identity matrix; the right side will now be A⁻¹.

Conclusion

• Understanding matrix multiplication and inversion is critical in linear algebra applications, including solving systems of equations.
• Different multiplication methods provide flexibility in calculations, while matrix inversion is essential for finding solutions to linear systems.