Math 166, Fall 2011,cBenjamin Aurispa5.1 Introduction to MatricesReminder: A matrix with m rows and n columns has size m x n. (This is also sometimes referred to as theorder of the matrix.) The entry in the ith row and jth column of a matrix A is denoted by aij.If the number of rows equals the number of columns, we call the matrix a .Consider the matrix: A =1 7 3 −39 1 −2 11−4 6 2 8What is the size of A? What is a23? What is a12?Two matrices are equal if they have the same size and all their corresponding entries are equal.We can add or subtract two matrices only if they have the same size. If they are the same size, we addor subtract them by adding or subtracting all their corresponding entries.If we multiply a matrix by a scalar (constant), then every entry in the matrix is multiplied by this scalar.Multiplying by a scalar does NOT change the size of the matrix.Example: Consider the matrices A =3 12 4−4 0and B =2 6−1 35 −2. Compute −2A + 3B.If A is an m x n matrix with entries aij, the transpose of A, denoted AT, is the n x m matrix found bymaking the rows of A the columns of AT.Example: Let A =1 −4−2 83 5. What is AT?Solve for the variables in the following matrix equation.2"a + 3 42 6#−"−1 69 7b − 1#="3 dc −8#T1Math 166, Fall 2011,cBenjamin AurispaMatrices are often used to organize and work with data, not solely for solving systems of equations.Example: The number of science and non-science majors enrolled in Math 131, 151, and 166 were countedduring two semesters. Matrix F below gives data for the fall semester and matrix S gives data for the springsemester.F = M131 M151 M166S 248 492 324NS 124 224 312!S = M131 M151 M166S 210 298 124NS 320 258 110!Find a matrix that gives the total number of science majors and the total number of non-science majorsenrolled in each of these classes over the 1-year period.Find a matrix that gives the average number of science majors and the average number of non-science majorsenrolled in each of these classes in a semester over this 1-year period.5.2 Matrix MultiplicationMultiplying matrices is not as easy as adding and subtracting them.If A is an m x n matrix and B is an r x s matrix, then for the matrix product AB to make sense, we musthave that n = r.In other words, the number of columns in A MUST equal the number of rows in B.The resulting matrix, AB, is an m x s matrix.2Math 166, Fall 2011,cBenjamin AurispaTo multiply AB, we move across the rows of A as we move down the columns of B. We multiplythe corresponding entries and add them up.The entry in row 1, column 1 of AB is found by using row 1 of A and column 1 of B.The entry in row 1, column 2 of AB is found by using row 1 of A and column 2 of B.The entry in row 2, column 1 of AB is found by using row 2 of A and column 1 of B.In general, if C = AB, then cij[row i, column j] is found by using row i of A and column j of B.Example: Let A ="2 1 −13 −2 a#and let B =−3 b1 −45 −1. What is AB?Example: Let A ="7 1 −3−9 8 4#and let B ="−3 2 41 −4 −6#. What is AB?You need to know how to multiply matrices by hand, but you can also use the calculator.Is it true that AB = BA? NO, matrix multiplication is NOT commutative.Let A ="1 2−3 5#and let B ="3 −11 4#. Find AB and BA.3Math 166, Fall 2011,cBenjamin AurispaMatrix multiplication can also be used in word problems. When multiplying matrices where the matriceshave meaning, you need to keep two things in mind:1. Size: Make sure the sizes of the matrices allow you to multiply them.2. Meaning: Label your matrices, and then make sure the labels on the columns of the first matrix matchthe labels on the rows of the second matrix. Otherwise, the multiplication doesn’t make sense.The number of children, students, and adults who attended movie I, II, and III on a given Friday are givenin the matrix A below. The admission price is $2 for children, $3 for students, and $4 for adults.A =C S AIIIIII225 110 5075 180 225280 85 110Come up with a matrix B to multiply A by where the product matrix will give the amount of money broughtin by each movie on that Friday.There are 4 possibilities:• B = and multiply• B = and multiply• B = and multiply• B = and multiply4Math 166, Fall 2011,cBenjamin AurispaExample: The matrix P below gives the number of units of rice, beef, and lamb per serving of three differenttypes of dog food. Matrix Q gives the number of servings of each type of dog food that are fed to threelarge dogs: Brutus, Max, and Atom.P =rice beef lambF ood I 5 9 12F ood II 4 8 3F ood III 7 10 11Q =F ood I F ood II F ood IIIBrutus 3 5 6Max 4 1 8Atom 7 2 3What is the meaning of the product P Q?What is the meaning of the product QP ?The Identity Matrix, Inis the n x n matrix that has 1’s all along the diagonal and 0’s everywhere else.For example, I2="1 00 1#and I3=1 0 00 1 00 0 1. I4=The identity matrix for matrices acts like the number 1 for numbers.We know that for any real number a, 1 · a = a and a · 1 = a.For any matrix A, InA = A and AIn= A (assuming the identity matrix is the appropriate size).5Math 166, Fall 2011,cBenjamin Aurispa5.3 Inverse of a Square MatrixLet A be a square matrix of size n.The inverse of A, denoted A−1, is the matrix such that A−1A = AA−1= InVerify that the following two matrices are inverses of each other: A ="4 52 3#and B ="32−52−1 2#AB = BA =Not all matrices have inverses.Square matrices that have inverses are said to be nonsingular.Matrices that do NOT have inverses are said to be singular.If a matrix is not square, it CANNOT have an inverse. If a matrix is square, it may or may not have aninverse.For this class, we will find matrix inverses on the calculator. Just call up the matrix and then press x−1.Find the inverse, if it exists, of each of the following matrices.A =1 −1 32 1 2−2 −2 1; A−1= B =1 2 0−3 4 −2−5 0 −2; B−1=We can use inverses to solve matrix equations.Solve the matrix equation AX = B for X.Solve the matrix equation XB − E = D for X.6Math 166, Fall 2011,cBenjamin AurispaSolve the matrix equation HX + X = F for X.If time allows:There is another way to solve systems of equations using matrices. It involves writing the system as amatrix equation AX = B.Example: Write the following system of linear equations as