Matrices

Description

A-Levels Further Mathematics Mind Map on Matrices, created by Alex Burden on 10/04/2017.
Alex Burden
Mind Map by Alex Burden, updated more than 1 year ago
Alex Burden
Created by Alex Burden over 7 years ago
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Resource summary

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

    Annotations:

    • m=n ⇒ square matrix
    1. Annotations:

      • 2x2 matrix
      1. Annotations:

        • 2x3 matrix
        1. Annotations:

          • 3x3 matrix
        2. Addition and Subtraction
          1. Annotations:

            • Matrices can only be added/subtracted if they are the same size
          2. Multiplication
            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)
            2. Identity Matrix
              1. The square matrix: AI=IA=A

                Annotations:

                • Acts like the number 1 in normal arithmatic
                1. Null Matrix
                  1. The square matrix such that all terms are 0
                    1. AB=0 does not imply that A=0 or B=0
                  2. Diagonal Matrix
                    1. A square matrix where all non-leading diagonal terms are 0
                      1. Determinants
                        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. |A|=ad-bc
                              1. |A|=a(ei-fh)-b(di-fg)+c(dh-eg)
                          1. Cofactors
                            1. Used to find the Determinant and Adjucate Matrix. Each term is either positive or negative depending on its position
                              1. Inverse Matrices
                                1. Transpose
                                  1. Matrix with all rows and columns interchanged

                                    Annotations:

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

                                        Annotations:

                                        • This is needed to find the inverse
                                        1. 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!
                                            1. 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
                                              2. 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

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