Understanding Matrices in College Algebra

Solve Systems with Matrices College Algebra

Describing Matrices A matrix is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Matrices are enclosed in are usually named with capital letters. Examples: 1 0 7 or , and 2 7 6 2 1 0 3 3 2 1 ? =1 2 4, ? = , ? = 5 8 3

Describing Matrices A matrix is often referred to by its size or dimensions: ? ? indicating ? rows and ? columns. Matrix entries are defined first by row and then by column. For example, the entry ???is located at row ?, column ?. ?11 ?21 ?31 A square matrix has dimensions ? ?, meaning it has the same number of rows and columns. A row matrix has dimensions 1 ? (one row), such as ? = ?11 ?12 ?22 ?32 ?13 ?23 ?33 ? = ?12 ?13 ?11 ?21 A column matrix has dimensions ? 1 (one column), such as ? =

Adding and Subtracting Matrices Given matrices ? and ? of like dimensions, addition and subtraction of ? and ? will produce matrix ? or matrix ? of the same dimension. ? + ? = ? such that ???+ ???= ??? ? ? = ? such that ??? ???= ??? Example: ? = 2 0 1 ? + ? =6 ? ? = 10 3 and ? =8 1 4 5 4 5 5 2 5 3

Multiplying a Matrix by a Scalar Scalar multiplication involves finding the product of a constant by each entry in the matrix. ?11 ?12 ?21 Scalar multiplication is distributive. For the matrices ? and ? with scalars ? and ?, ? ? + ? = ?? + ?? ? + ? ? = ?? + ?? ??11 ??21 ??12 ??22 Given ? = ?22, the scalar multiple ?? =

Product of Two Matrices The product of two matrices is only possible when the inner dimensions are the same. If ? is an ? ? matrix and ? is an ? ? matrix, the product matrix ?? is an ? ? matrix. To obtain the entry in row ?, column ? of ??, multiply row ? in ? by column ? in ? as follows: ?1? ?2? ??? Matrix multiplication is associative: ?? ? = ?(??) Matrix multiplication is distributive: ? ? + ? = ?? + ?? ??1 ??2 ??? = ??1?1?+ ??2?2?+ + ??????

Product of Two Matrices 7 8 9 10 11 12 Example: Multiply ? =1 2 5 3 6 and ? = 4 Solution: Since ? has dimension 2 3 and ? has dimension 3 2, ?? has dimension 2 2. 1 7 + 2 8 + 3(9) 4 7 + 5 8 + 6(9) 50 122 1 10 + 2 11 + 3(12) 4(10) + 5 11 + 6(12) 68 167 ?? = ?? =

Systems of Equations and Matrices A matrix can be used to represent and solve a system of equations. The coefficients of the variables and the constants become the entries in a matrix. A vertical line separates the coefficient entries from the constants; this is called an augmented matrix. Example: 3? ? ? = 0 ? + ? = 5 2? 3? = 2 Notice that all the variables line up in their own columns, and missing terms have a coefficient of 0. 3 1 2 1 1 0 1 0 3 0 5 2 is represented as |

Row Operations and Row-Echelon Form In order to solve the system of equations, we convert the augmented matrix to row-echelon form in which there are ones down the main diagonal, and zeros in every position below the main diagonal. 1 ? 0 1 0 0 We use row operations to obtain a new matrix that is row-equivalent. 1. Interchange rows. (Notation: ?? 2. Multiply a row by a constant. (Notation: ???) 3. Add the product of a row multiplied by a constant to another row. (Notation: ??+ ???) ? ? 1 ??)

Gaussian Elimination The Gaussian elimination method refers to a strategy to obtain the reduced row-echelon form of a matrix. Example: 1 3 2 1 2 0 32 1 8?2= ?2 0 4 1 0 0 4 2 2|10 10 Obtain a zero in row 2, column 1 3?1+ ?2= ?2 8| 2 1|10 1|2 1 Obtain a one in row 2, column 2 Obtain a zero in row 1, column 2 2?2+ ?1= ?1

Gaussian Elimination Given a system of equations, create the augmented matrix from the coefficients and constants. Reduce the matrix to row-echelon form to obtain the solution for the system. For example: ? + 2? = 10 3? 2? = 2is written as 1 2 2|10 3 2 Reduce this matrix to 1 0 1|2 0 4 Rewrite the system as ? = 2 ? = 4for the solution of 2,4

Identity Matrix and Multiplicative Inverse The identity matrix, ??, is a square matrix containing ones down the main diagonal and zeros everywhere else. 1 0 0 0 1 0 0 0 1 ?2=1 0 1 ?3= 0 If ? is an ? ? matrix and ? is an ? ? matrix such that ?? = ?? = ??, then ? = ? 1, the multiplicative inverse of matrix ?. 1 5 2 2 1 9 + 5(2) 2 9 9(2) 9, ? = 9 5 1 1 5 + 5(1) 2 5 9(1) Example: ? = =1 0 1 ?? = = ?2 0

Find the Inverse Using Matrix Multiplication To find the inverse of a given matrix, multiply it by a matrix containing unknown constants and set it equal to the identity. This will result in systems of equations that can be used to find the unknowns. Example: ? =1 2 3 1 2 2 ? ? 0 1 2 =1 0 3 ? ? Set up ? ? 1= ? =1 0 1 1? 2? 2? 3? 1? 2? 2? 3? Find the product 0 1? 2? = 1 2? 3? = 0, 1? 2? = 0 Create systems and solve for ?, ? and ?, ? 2? 3? = 1

Find the Inverse by Augmenting with the Identity When matrix ? is transformed into ?, the augmented matrix ? transforms into ? 1. Example: ? =2 5 3 2 1 5 0 1 1 3|1 0 Augment ? with the identity 1 0 0 1| 3 1 2 Perform row operations to turn ? into the identity 5 3 1 2 Inverse is the right side of the augmented matrix ? 1= 5

Inverse of a 22 Matrix If ? is a 2 2 matrix such as ? =? given by the formula ? ?, the multiplicative inverse of ? is ? 1 ? ? ? ? 1= ? ?? ?? If ?? ?? = 0, then ? has no inverse. Example: ? =1 2 3 2 3 2 2 1 = 3 2 1 1 Solution: ? 1= 2 1 3 ( 2)(2)

Solve a System of Equations Using an Inverse Given a system of equations, write the coefficient matrix ?, the variable matrix ?, and the constant matrix ?. Then ?? = ?. Multiply both sides by the inverse of ? to obtain the solution, ? = ? 1?. Example: Solve the system of equations 3? + 8? = 5 4? + 11? = 7 ? ?=5 3 4 8 Write the system in matrix terms 11 7 11 4 8 3 Use the formula for the inverse of a 2 2 matrix ? 1= ? ?= 11 5 + 8 7 4 5 + 3(7) 11 4 8 3 = 1 5 7 The solution is 1,1 = 1

Quick Review What are the dimensions of a matrix? How do you find the product of two matrices? What is an augmented matrix? What are the characteristics of a matrix in row-echelon form? What are the three row operations used to obtain row-echelon form? How do you find the inverse of a 2 2 matrix? Can the inverse be found for all dimensions of a matrix? How do you solve a system of equations using Gaussian elimination? How do you solve a system using the inverse of a matrix?