Matrices
Matrix - a rectangular array of variables or constants in horizontal rows and vertical columns enclosed in brackets. Element - each value in a matrix; either a number or a constant. Dimension - number of rows by number of columns of a matrix. **A matrix is named by its dimensions.
Examples: Find the dimensions of each matrix. Dimensions: 3x2Dimensions: 4x1 Dimensions: 2x4
Different types of Matrices Column Matrix - a matrix with only one column. Row Matrix - a matrix with only one row. Square Matrix - a matrix that has the same number of rows and columns.
Equal Matrices - two matrices that have the same dimensions and each element of one matrix is equal to the corresponding element of the other matrix. *The definition of equal matrices can be used to find values when elements of the matrices are algebraic expressions.
* Since the matrices are equal, the corresponding elements are equal! * Form two linear equations. * Solve the system using substitution. Examples: Find the values for x and y
Set each element equal and solve! 2.
Matrix Operations Addition Subtraction Multiplication Inverse
Addition
Addition Conformability To add two matrices A and B: # of rows in A = # of rows in B # of columns in A = # of columns in B
Subtraction
Subtraction Conformability To subtract two matrices A and B: # of rows in A = # of rows in B # of columns in A = # of columns in B
Multiplication Conformability Regular Multiplication To multiply two matrices A and B: # of columns in A = # of rows in B Multiply: A (m x n) by B (n by p)
Multiplication General Formula
Multiplication I
Multiplication II
Multiplication III
Multiplication IV
Multiplication V
Multiplication VI
Multiplication VII
Inner Product of a Vector (Column) Vector c (n x 1)
Outer Product of a Vector (Column) vector c (n x 1)
Inverse of 2 x 2 matrix Find the determinant = (a 11 x a 22 ) - (a 21 x a 12 ) For det(A) = (2x3) – (1x5) = 1
Inverse of 2 x 2 matrix Swap elements a 11 and a 22 Thus becomes
Inverse of 2 x 2 matrix Change sign of a 12 and a 21 Thus becomes
Inverse of 2 x 2 matrix Divide every element by the determinant Thus becomes (luckily the determinant was 1)
Inverse of 2 x 2 matrix Check results with A -1 A = I Thus equals