MATRICES Introduction Matrix algebra has several uses in

MATRICES

Introduction • Matrix algebra has several uses in economics as well as other fields of study. • One important application of Matrices is that it enables us to handle a large system of equations. • It also allows us to test for the existence of a solution to a system of equations even before we attempt solving them.

Uses of Matrices • A company with several outlets selling several different products, a matrix provides a concise way of keeping track of stock. oulet Milo Milk Bar Soap Bath Gel Ghana 120 110 90 150 Nigeria 300 180 210 110 Togo 175 190 160 80 Benin 140 170 180 140 • Reading across a row of the matrix, the firm (Nestle) can determine the level of stock in any of its outlets. • By reading down the column, the firm can determine the stock of any of its products.

Matrices: Definition and Terms • 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

Dimensions of Matrices Dimensions: 3 x 2 Dimensions: 4 x 1 Dimensions: 2 x 4

Diagonal matrix: Trace:

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. • Identity Matrix - An identity matrix is a square which has 1 for every element on the principal diagonal from left to right and 0 everywhere else. • 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.

* Write as linear equations. * Combine like terms. * Solve using elimination.

3. Set each element equal and solve!

Basic Matrix Operations: Transpose Elementary Linear Algebra: Section 2. 2, p. 57

(Find the transpose of the following matrix) (a) Sol: (a) (b) (c )

Properties of Transpose Matrices

Symmetric matrix: A square matrix A is symmetric if A = AT Skew-symmetric matrix: A square matrix A is skew-symmetric if AT = –A Ex: is symmetric, find a, b, c? Sol:

Matrix Algebra: Addition and Subtraction • If A and B are both m x n matrices, the sum of A and B is defined to be the m x n matrix A+B obtained by adding corresponding elements. • This means that for addition and subtraction of matrices, the matrices involved should have the same order.

Matrix addition: Ex 2: (Matrix addition)

Scalar multiplication: Matrix subtraction: Ex 3: (Scalar multiplication and matrix subtraction) Find (a) 3 A, (b) –B, (c) 3 A – B

Sol: (a) (b) (c)

Properties of matrix addition and scalar multiplication: Then (1) A+B = B + A (2) A + ( B + C ) = ( A + B ) + C (3) ( cd ) A = c ( d. A ) (4) 1 A = A (5) c( A+B ) = c. A + c. B (6) ( c+d ) A = c. A + d. A

Matrix Multiplication: • Consider the Matrices A and B • Scalar Multiplication: Find the following • 2 A • ½B • 3 B-A

Matric Multiplication: 2 Matrices •

**Multiply rows times columns. **You can only multiply if the number of columns in the 1 st matrix is equal to the number of rows in the 2 nd matrix. They must match. Dimensions: 2 x 3 3 x 2 The dimensions of your answer.

Examples: 2(3) + -1(5) 2(-9) + -1(7) 2(2) + -1(-6) 3(3) + 4(5) 3(-9) + 4(7) 3(2) + 4(-6)

Dimensions: 2 x 3 2 x 2 *They don’t match so can’t be multiplied together. *

Properties of matrix multiplication: (1) A(BC) = (AB)C (2) A(B+C) = AB + AC (3) (A+B)C = AC + BC (4) c (AB) = (c. A) B = A (c. B)

Commutative, Associative and Distributive Laws • Commutative: A + B = B + A • Associative: (A+B)+C = A(B+C) (AB)C= A(BC) • Distributive: A(=B+C)= AB+AC (A+B)+C = AC + AB

Trial Questions: Proofs Show whether or not the following relations are true given the following matrices A, B and C. • A (B+C) = AB +AC • (AB)C=A(BC)

Trial Questions: Addition and Subtraction