Matrices

Matrices

1. m x n is the size of the matrix with m rows and n columns

   • m=n ⇒ square matrix
   • • 2x2 matrix
   • • 2x3 matrix
   • • 3x3 matrix

2. Addition and Subtraction

   • • Matrices can only be added/subtracted if they are the same size

3. Multiplication

   • • 2x3 and 4x5 matrices cannot be multiplied Multiplying a 2x3 and 3x5 matrix together gives a 2x5 matrix AB ≠ BA The order of multiplication is very important ABC=(AB)C=A(BC)

4. Identity Matrix

   • The square matrix: AI=IA=A
     • Acts like the number 1 in normal arithmatic

5. Null Matrix

   • The square matrix such that all terms are 0
     • AB=0 does not imply that A=0 or B=0

6. Diagonal Matrix

   • A square matrix where all non-leading diagonal terms are 0

7. Determinants

   • This is required to find the Inverse of a matrix
     • A Matrix is singular when the determinant is 0 Is determined by |A| or det|A|
     • • |A|=ad-bc
     • • |A|=a(ei-fh)-b(di-fg)+c(dh-eg)

8. Cofactors

   • Used to find the Determinant and Adjucate Matrix. Each term is either positive or negative depending on its position

9. Inverse Matrices

   • Transpose
     • Matrix with all rows and columns interchanged
       • The leading diagonal is unchanged!
   • Adjucate
     • The adjucate matrix of A is the transpose matrix of the cofactors of A.
       • This is needed to find the inverse
   • Inverse
     • A is a non-singular matrix, then the inverse is defined; A^-1A=I=AA^-1
       • If the determinant is 0, then there is no inverse!
   • Inverse and Multiplying
     • AB=C ⇒ B=A(^-1)C
       • When multipying, the order is important! A(^-1)AB=B but ABA(^-1)≠B
   • Solution of Linear Equations
     • To solve a system of linear equations; Express in the form AB=C where B= Find the inverse matrix A^-1. Solution given by B= =A(^-1)C
       • This method can only be used if |A|≠0
       • A= B= C=
         • AB=C ⬄ x = ⇒
           • BUT B=A(^-1)C ⇒ = A^-1

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