Matrices 1

Matrices 1

1. A matrix is an array of elements set out in a pair of brackets.

2. M = (1 2) is a matrix, it has 2 elements and its dimensions are 1x2

   • Dimensions of Matrices are always written as Rows x Columns

3. Adding and Subtracting Matrices

   • We can only add or subtract Matrices that have the same dimensions
   • Just add or subtract the corresponding elements in each matrix (r1,c1 + r1,c1)

4. Multiplication of matrices

   • To multiply a matrix by a scalar multiply each element in the matrix by the scalar (as if it were a normal set of brackets)
   • Only conformable matrices can be multiplied
   • Conformable: Number of columns in Matrix 1 = Number of rows in Matrix 2

5. Matrices are non-commutitive (Order matters)

   • B(A) = "Pre Multiply by B"
   • (A)B = "Post Multiply by B"
   • (AB) Is NOT equal to (BA)

6. Matrices are Associative e.g. ( (BC)A = B(CA))

7. The Identity Matrix

   • A Square Matrix
     • Must be square otherwise the dimension of M may change
   • All elements on the leading diagonal are 1, everything else is 0
   • MI = M

8. Division of Matrices

   • There is no division within Matrix Algebra does not exist so we multiply by an inverse

9. Determinant

   • The determinant is the elements on the leading diagonal (ad) - the elements on the remaining diagonal multiplied (bc)
   • Det(M) = AD-BC
   • Can be found for any square matrix
   • If Det(M) = 0 M is singular, If not it is non-Singular

10. A real number is any number that can be placed on a number line

11. We can find the inverse of any square matrix that is non-singular

    • Singular Matrices have no inverse

12. The inverse of a 2x2 matrix is 1/Det(M) x A where A is M where the elements in the leading diagonal are swapped and the elements on the remaining diagonal are multiplied by -1

13. If the original Matrix is A then the transpose matrix is A(To the power of T)

    • If A= A^T, A is a symmetric Matrix
      • Any symmetric Matrix is square
    • A^T is when the leading diagonal stays the same all other elements are reflected in the leading diagonal

14. (AB)^T =B^T x A^T

15. (AB)^-1 = B^-1 x A^-1

16. Determinant of a 3x3 Matrix

    • Step 1: Extend the Matrix, Step 2: Cover row 1, Step 3: Cover Column 1, Step 4: Find the number under intersection, Step 5: Find the determinant of the 2x2 matrix, Step 6:Repeat covering up the next column.
    • Inverse of a 3x3 Matrix
      • M^-1 = 1/Det(M) x C^T
        • (A B C) (D E F) (G H I)
          • a,b& c are the determinants of the 2x2 matrices where row 1 is covered
          • d,e & f are the deteminants of the 2x2 matrices where row 2 is covered
          • g.h & i are the determinants of the 2x2 matrices where row 3 is covered
          • Then reflect C in the Leading diagonal and x by 1/Det(M)

17. When M is singular (Det(M) = 0) the inverse does not exist, graphically the lines are parallel. There are no solutions

18. In 2D Space, ax + by = c

19. In 3D Space, ax + by + cz = d

20. Is Det(M) = 0

    • No, M^-1 exists, 3 planes meet at a single point.
    • Yes, M^-1 Doesnt exist, Either a Sheaf Or a Prism
      • Sheaf - Line of intersection between 3 planes, an infinite number of solutions along this line. The system of the equations is consistent
      • Prism - No solution satisfy all 3 planes. The System of equations is inconsistent