1.5 Matricies

A matrix is a rectangular array of numbers arranged in M horizontal rows and N vertical columns.   A is a 2x3 matrix 2 rows 3 columns

Location of an array element is noted by row and column. a11 indicates row 1 column 1 a12 indicates row 1 column 2 a13 indicates row 1 column 3 a21 indicates row 2 column 1 a22 indicates row 2 column 2 a23 indicates row 2 column 3

A = a12 = 3 row 1 column 2

Matrix Rules Addition Can only add matrices that have that have the same number of rows and columns. A = + B= = =

The result of the multiplication are the circled numbers. Can only multiply matrices if the number of columns in the first matrix is equal to the number of rows in the second matrix. row x col A is a 2 x 3 matrix B is a 3 x 2 matrix The multiplication result is the row of the first matrix by the column of the second matrix, in this example, a 2 x 2. A = x B = x Check if the numbers in the squares are the same, if so, you can multiply these matrices. The result of the multiplication are the circled numbers. 2 3 3 2

Multiply row of A to column of B A = X B = =

A matrix is square if it has the same number of rows as columns A zero matrix is a matrix with entries that are all zero.

Transpose A is a 2X3 matrix A= Transpose A into a 3X2 matrix AT =

Main diagonal Top left to bottom right

Symmetric matrix A matrix is called symmetric if it is square and equal to its transpose. A= AT

Skew Symmetric Skew symmetric is where the main diagonal is the same as the transpose main diagonal and all other numbers are opposite in location. A = AT

Boolean Matrix Operations (1’s and 0’s) ∨ means OR ∧ means AND A ∨ B = 1 if A or b, or both A and B = 1 A ∨ B = 0 only when both A and B =0 A ∨ B = C A = B= C=

A ∧ B =1 only when A = 1 and B = 1 A ∧ B = C A = B= C=

Boolean Product Can multiply matrices if the number of columns of the first matrix is equal to the number of rows in the second matrix. Boolean product of A and B is denoted A ⊙ B row X col row X col A is a 2 x 2 B is a 2 X 3 Number of columns of A = Number of rows of B. Result will be a 2 X 3 (row of first matrix, column of second matrix) A = ⊙ B= = =