Matrices

Annotations:

• m=n ⇒ square matrix

1. Annotations:

   • 2x2 matrix

2. Annotations:

   • 2x3 matrix

3. Annotations:

   • 3x3 matrix

1. Annotations:

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

1. Annotations:

   • 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)

1. The square matrix: AI=IA=A

   Annotations:

   • Acts like the number 1 in normal arithmatic

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

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

1. This is required to find the Inverse of a matrix

   Annotations:

   • A Matrix is singular when the determinant is 0 Is determined by |A| or det|A|
   1. 1. |A|=ad-bc
   2. 1. |A|=a(ei-fh)-b(di-fg)+c(dh-eg)

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

1. Transpose
   1. Matrix with all rows and columns interchanged

      Annotations:

      • The leading diagonal is unchanged!
2. Adjucate
   1. The adjucate matrix of A is the transpose matrix of the cofactors of A.

      Annotations:

      • This is needed to find the inverse
3. Inverse
   1. A is a non-singular matrix, then the inverse is defined; A^-1A=I=AA^-1

      Annotations:

      • If the determinant is 0, then there is no inverse!
4. Inverse and Multiplying
   1. AB=C ⇒ B=A(^-1)C

      Annotations:

      • When multipying, the order is important! A(^-1)AB=B but ABA(^-1)≠B
5. Solution of Linear Equations
   1. 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

      Annotations:

      • This method can only be used if |A|≠0
      1. A= B= C=
         1. AB=C ⬄ x = ⇒
            1. BUT B=A(^-1)C ⇒ = A^-1