Matrices

Element - each value in a matrix; either a number or a constant. 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: 3x2 Dimensions: 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.

Examples: Find the values for x and y * Since the matrices are equal, the corresponding elements are equal! * Form two linear equations. * Solve the system using substitution.

Set each element equal and solve! 2. Set each element equal and solve!

Matrix Operations Addition Subtraction Multiplication Inverse

Addition

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

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 = (a11 x a22) - (a21 x a12) For det(A) = (2x3) – (1x5) = 1

Inverse of 2 x 2 matrix Swap elements a11 and a22 Thus becomes

Inverse of 2 x 2 matrix Change sign of a12 and a21 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