Operations with Matrices Big Idea Organizing data into a matrix can make analysis and interpretation much easier. Operations such as addition, subtraction, and scalar multiplication can be performed on large sets of data in matrix form. To add or subtract two matrices, the matrices must have the same dimensions. Then, calculate the sum or difference of the corresponding entries. The resulting matrix will have the same dimensions as original matrices being combined. Scalar multiplication is done by distributing the scalar to every element within the matrix being multiplied. The result is a product with same dimensions as the original matrix. Objectives: A.CED.3 A. Create equations that describe numbers or relationships - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Skills Analyze data in matrices Perform algebraic operations with matrices Vocabulary Scalar - A constant. Scalar multiplication - Multiplying any matrix by a scalar. Algebra 2 Unit 03d Operations on Matrices rev Page of 9 4/7/204
Examples, Notes, and Exam Questions Matrix Operations Goal: Learn to add and subtract matrices, use scalar multiplication, and solve a matrix equation A matrix is a rectangular arraignment of numbers in rows and columns. For example a matrix A with nmdimensions has n number of rows and m number of columns. The values within the matrix are known as the entries and are indentified by their position in the matrix. Ex : Determine the dimensions of matrix A and identify the position of each entry. 4 3 0 2 A 2 7 9 Special Types of Matrices 2 A matrix with one row is defined as a row vector. Example: 3 4 0 3 5 A matrix with one column is defined as a column vector. Example: 5 0 A matrix with the same number of rows as columns is defined as square matrix. Example: 3 7 4 0 A matrix with all zero entries is defined as a zero matrix. Example: 0 0 0 0 Equal Matrices Two matrices are equal if their dimensions are the same and the corresponding entries are equal. 4 2 Example: Matrix A = Matrix B 4 2 A ; B.5 0 0 2 Ex 2: : Determine if the following matrices are equal. 3.5 3 A 2; B 6.5 4 2 Algebra 2 Unit 03d Operations on Matrices rev Page 2 of 9 4/7/204
Adding & Subtracting Matrices WARNING!!! In order to add or subtract the entries of two matrices, they must have the same dimensions. Adding and subtracting are done by adding or subtracting the corresponding entries of each matrix. 7 3 Ex 3: Add 2 2 0 Ex 4: Subtract 3 5 0 7 0 2 2 3 3 0 5 6 5 2 2 A non-entry real number is defined as a scalar. It is typically a value that is distributed into a matrix with multiplication. Ex 5: Multiply 3Aif 2 0 A 3 Solving Matrix Equations You can use matrix and their equality concepts to solve matrix equation. Ex 6: Solve the matrix for x and y: 8 0 4 2x 48 48 4 2y 6 0 8 Ex 7: Solve the matrix for x and y: 0 2 x 5 0 2 3 5 4y 2 9 Algebra 2 Unit 03d Operations on Matrices rev Page 3 of 9 4/7/204
. To add or subtract to matrices, what must be true? SAMPLE QUESTIONS 2. Perform the indicated operation(s), if possible. 0 8 8 2 a. 4 0 5 2 9 b. 3 2 4 2 3 3 0 0 2 3. Tell whether the matrices are equal or not equal. 3 2 7 2.5 4.25 2 a. ; 4 0.5 0.5 0 2 0 2 4. Perform the indicated operation, if possible. If not possible, state the reason. 3 2 3 5 2 a. 3 0 2 b. 2 2 4 5 9 3 2 5. Perform the indicated operation. 0 0 a. 9 2 2 4 3 9 b. 6. Perform the indicated operation. 7 0 4 7 a. 2 2 8 6 2 3 5 5 7. Solve the matrix equation for x and y. 3x 2 4 0 6 2 a. 8 7 8 y 0 Algebra 2 Unit 03d Operations on Matrices rev Page 4 of 9 4/7/204
8. Use matrices to organize the following information about car insurance rates. This year: For car, Comprehensive, collision, and basic insurance cost $62.5, $58.29, and $486.9. For 2 cars, Comprehensive, collision, and basic insurance cost $50.32, $984.6, and $892.5. Next year: For car, Comprehensive, collision, and basic insurance cost $66.28, $520.39, and $490.05. For 2 cars, Comprehensive, collision, and basic insurance cost $55.84, $987.82, and $895.3. Algebra 2 Unit 03d Operations on Matrices rev Page 5 of 9 4/7/204
Multiplying Matrices Big Idea Matrix multiplication can be performed if the number of columns in the first matrix is equal to the number of rows in the second matrix. This is done by multiplying the elements in the first row by the corresponding elements of the second matrices columns. The resulting products are combined to form a single entry. The matrix dimensions will consist of the number of rows in the first matrix by the number of columns in the second matrix. Objectives: A.CED.3 A. Create equations that describe numbers or relationships - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Skills Multiply matrices Use the properties of matrix multiplication Vocabulary N/A Algebra 2 Unit 03d Operations on Matrices rev Page 6 of 9 4/7/204
Examples, Notes, and Exam Questions Multiplying Matrices Goal: Learn to multiply two matrices Remember: In order to add or subtract two matrices they must have the same. 2 4 0 2 2 0 3 2 Ex 8: Subtract 9 0 2 4 4 9 4 dimension. WARNING!!! To multiply two matrices A and B, the number of columns in A must equal the number of rows in B. Example: if A is a mnmatrix, then B must be an n p, and the product results in a m pmatrix. 2 0 If A 2 5 2 3 and B 3 0, then 45 AB 0 3 28 20 Multiplication Process for Matrices 2 0 Ex 9: : 2 4 3 4 st. Multiply the corresponding entries of row one with those of column one in the second matrix and sum the products into a single solution. 2 nd. The product sum from step one is the first entry of your solution matrix. 3 rd. Repeat steps and 2 until each row of the first matrix has been multiplied by each column of the second matrix. Ex 0: 5 4 3 5 2 6 8 0 4 Algebra 2 Unit 03d Operations on Matrices rev Page 7 of 9 4/7/204
Ex : Let WARNING!!! Matrix multiplication is NOT commutative. A 4 4 3 0 2 and B 2, then find AB and BA. 2 2 0 0 3 Ex 2: If A, B,& C 4 3 2 2, find B( A C) and BA BC. Algebra 2 Unit 03d Operations on Matrices rev Page 8 of 9 4/7/204
SAMPLE QUESTIONS. Complete this statement: The product of matrices A and B is defined provided the number of in A is equal to the number of in B. 2. State whether the product of AB is defined. If so, give the dimensions of AB. a. A 33; B 3 4 3. Find the product. 2 a. 4 4 3 4. Two lacrosse teams submit equipment lists to their sponsors. Women s team: 5 sticks, 5 balls, and 6 uniforms. Men s team: 8 sticks, 22 balls, and 7 uniforms. Each stick costs $55, each ball costs $6, and each uniform costs $35. Use matrix multiplication to find the total cost of the equipment for each team. 5. State whether the product AB is defined. If so, give the dimensions of AB. a. A: 24, B: 4 3 b. A: 24, B:3 4 6. Find the product. If it is not defined, state the reason. 0 0 5 7 4 a. 6 3 3 2 6 2 5 3 4 5 2 7. Use the given matrices, simplify the expression. 3 2 2 5 6 4 2 0 3 A ; B ; C ; D 2 4 ; E 4 2 6 2 4 2 2 3 3 3 4 a. 2AB b. E D E Algebra 2 Unit 03d Operations on Matrices rev Page 9 of 9 4/7/204
8. Solve for x and y. 2 2 6 a. 3 2 4 x 9 0 2 4 3 y A Algebra 2 Unit 03d Operations on Matrices rev Page 0 of 9 4/7/204
Solving Systems of Equations using Cramer's Rule Big Idea Systems of equations can be written as a matrix. In this form Cramer's Rule can be used to solve the system. Cramer's Rule utilizes the determinant which can be found by calculating the difference of the downward diagonals with upward diagonals. It is important to note that a system as no solution if the determinant in the denominator is zero. Objectives: A.CED.3 A. Create equations that describe numbers or relationships - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Skills Evaluate determinants Solve system of linear equations by using Cramer's Rule Vocabulary Determinant - A square array of numbers or variables enclosed between two parallel lines. Second-order determinant - The determinant of a 22matrix. Third-order determinant - The determinant of a 33matrix. Diagonal rule - A method for finding the determinant of a 33matrix. Cramer's Rule - A method that uses determinants to solve a system of equations. Coefficient matrix - A matrix that contains only coefficients of a system of equations. Algebra 2 Unit 03d Operations on Matrices rev Page of 9 4/7/204
Examples, Notes, and Exam Questions Determinants and Cramer s Rule Goal: Learn to evaluate the determinant of a 22&33matrix, and then use Cramer s Rule to solve a system of equations. Systems of equations can be solved by graphing, substitution, or elimination (combination). All methods define the solution of the system. That is, the input value (x) that produces an equal output value (y) for both functions, and the point where the lines intersect. Solve the following system: 2 x y 3x 2y 23 Determinants and Cramer s Rule provide a matrix approach to solving linear systems. A Determinant of a 22matrix is the difference of the products of the entries on the diagonals. The determinant of a matrix A is denoted by det A or A. a Example: Given A c b d, find A. Solution: det A A ad c b Example: Evaluate the determinant of the two by two matrix: A 2 4 3 The determinant of a 33matrix is the difference between the sums of the products of the entries on the diagonals. a b c Example: Given B d e f, finddet B. Solution: g h i det B aei b f g cd h g ec h f a id b Example: Evaluate the determinant of the 33matrix: 4 3 A 5 7 0 2 2 Determinants can be used to find the area of a triangle whose vertices are points in a coordinate plane. The area of a triangle in a coordinate plane can be found using: Algebra 2 Unit 03d Operations on Matrices rev Page 2 of 9 4/7/204
x y Area x y 2 x y 2 2 3 3, the ensures a positive area. Example: Find the area of a triangle with vertices at 2, 2, 2,4,& 5,. Cramer s Rule is a matrix method for solving linear systems, which utilizes the coefficient matrix and determinants. Determining the coefficient matrix ax by e Given a linear system cx dy f, the coefficient matrix is defined by the coefficients of x and y; a b c d. Example: Determine the coefficient matrix for the linear system 2 x y and then write as a matrix 3x 2y 23 equation. Steps to solve a linear system using Cramer s Rule st. Determine the coefficient matrix Example: 4 x 6 y 4 x5y 4 2 nd. Calculate the determinant of the coefficient matrix found is step. 3 rd. Substitute the constant values in for the x coefficients of the coefficient matrix and calculate this determinant. 4 th. Substitute the constant values in for the y coefficients of the coefficient matrix and calculate the determinant. 5 th. Divide the determinant values found in steps 3 and 4 by the determinant value of the coefficient matrix found in step 2. Note: If the determinant of the coefficient matrix does NOT equal zero, then the system has exactly one solution. Example: Solve 8 x 5 y 2 2x 4y 0 Algebra 2 Unit 03d Operations on Matrices rev Page 3 of 9 4/7/204
SAMPLE QUESTIONS. Evaluate the determinant of the matrix. 3 0 a. 2 2 b. 2 4 2 2. Evaluate the determinant of the matrix. 3 2 a. 0 9 0 5 6 2 3. Use Cramer s rule to solve the linear system. 6x8y 4 a. 4x 5y 4 b. 2 x 7 y 3 3x8y 23 4. What is the determinant of A. 7 B. 2 C. D. 2 3 2? 5. Cramer s Rule is used to solve the system of equations below. 4x 5 y z 3x 2 y 2z 5 2x 6 y 3z 8 Which determinant represents the denominator for the solution of z? 4 5 E. 5 3 2 8 2 6 4 5 F. 3 3 2 2 8 6 Algebra 2 Unit 03d Operations on Matrices rev Page 4 of 9 4/7/204
G. H. 4 5 3 2 5 2 6 8 4 5 3 2 2 2 6 3 Algebra 2 Unit 03d Operations on Matrices rev Page 5 of 9 4/7/204
Solving Systems of Equations Using Inverse Matrices (Section 8) Big Idea The identity matrix is a square matrix with diagonal entries of and 0 for all other entries. The product of a matrix and it's inverse is the identity matrix. Thus, an inverse matrix can be used to solve a system of equations written in matrix form. An inverse matrix is found by the product of two factors. The first of which is one of the determinant of the coefficient matrix and the other is the modified coefficient matrix where the entries of the downward diagonals are switched and the upward diagonal entries are negated. Objectives: A.CED.3 A. Create equations that describe numbers or relationships - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Skills Find the inverse of a 22matrix Write and solve matrix equations for a system of equations Vocabulary Identity matrix - A square matrix that, when multiplied by another matrix, equals that same matrix. If A is any nnmatrix and I is the nnidentity matrix, then AI Aand I A A. Square matrix - A matrix with the same number of rows as columns. Inverse matrix - Two nnmatrices are inverses of each other their product is the identity matrix. Matrix equation - A matrix form used to represent a system of equations. Variable matrix - A matrix that only contains the variables of a system of equations. Constant matrix - A matrix that only contains the constants of a system of equations. Algebra 2 Unit 03d Operations on Matrices rev Page 6 of 9 4/7/204
Examples, Notes, and Exam Questions Goal: Learn to identify and use inverse matrices to solve linear systems of equations. Remember: To find the determinant of a 22matrix we take the difference of the products of the diagonal entries. 2 4 Example: Evaluate the determinant of A 3 Additionally, we should recall that the Inverse property states; for any real number a 0, a a where is the multiplicative identity. We use this concept of Multiplicative Inverse to solve problems such as 4x 2 for x, by multiplying both sides of the equation by the inverse of 4. Two nnmatrices are inverses of each other if their product equals the nnidentity matrix. An n n identity matrix has ones on the downward diagonal and zeros for every other entry. Example: 22Identity Matrix: 33Identity Matrix: The inverse matrix can be used to solve matrix equations and is denoted by A. The inverse of a matrix a b d b d b A c d is; A, ad bc 0 A c a ad bc c a Steps to find the inverse of a matrix st. Find the determinant of A. Example: Find the inverse of A 3 2 2 4 2 nd. Put matrix A in inverse form, that is switch the a and d entries and negate the b and c entries. 3 rd. Distribute the ratio found in step into the matrix formed in step 2. Steps to solve a matrix equation with the inverse matrix 3 4 3 8 st. Determine the matrix equation AX B Example: X 5 7 2 2 2 nd. Find the inverse A. Algebra 2 Unit 03d Operations on Matrices rev Page 7 of 9 4/7/204
3 rd. Multiply both sides of the equation by A. Be sure to multiply to the left of each matrix. That is, A AX A B Inverse matrices can be used to solve a system of equations Steps to solving a linear system of equations with inverse matrices 3x 4y 5 st. Write the system of equations as a Example: Solve 2x y 0 matrix equation. AX B, where A is the coefficient matrix and B is the constant matrix 2 nd. Find the inverse A of the coefficient matrix. 3 rd. Multiply both sides of the matrix equation by A. Algebra 2 Unit 03d Operations on Matrices rev Page 8 of 9 4/7/204
SAMPLE QUESTIONS. Find the inverse of the matrix. 5 4 a. 4 4 b. 3 2 2 2 2. Solve the matrix equation. 5 3 3 a. X 0 5 4 0 b. 3 7 8 5 7 3 X 4 5 2 9 3. Write the linear system as a matrix equation. x3z 6 2x 3y z 3x y 2z 3 4. Use an inverse matrix to solve the linear system. x 2y 3 2x8y 5. Write the linear system as a matrix equation. 2x 5y 3x 7y5 6. Use an inverse matrix to solve the linear system. 5x 7y 9 2x3y3 7. Solve the matrix equation 7 5 8 2 9 3 X 4 3 6 6 2. QOD: What is the special relationship between a matrix and its inverse? Algebra 2 Unit 03d Operations on Matrices rev Page 9 of 9 4/7/204