Math 313 Chapter 1 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 1.1 Introduction to Systems of Linear Equations 2 2 1.2 Gaussian Elimination 3 3 1.3 Matrices and Matrix Operations 5 4 1.4 Inverses; Rules of Matrix Arithmetic 6 5 1.5 Elementary Matrices and a Method for Finding A 1 7 6 1.6 Further Results on Systems of Equations and Invertibility 9 7 1.7 Diagonal, Triangular, and Symmetric Matrices 11 1
1 1.1 Introduction to Systems of Linear Equations Important Formulas to Remember Elementary Row Operations 1. Multiply a row through by a nonzero constant 2. Interchange two rows 3. Add a multiple of one row to another row Identify whether or not the following equations are linear. (a) x 1 + 5x 2 2x 3 = 1 (b) x 1 + 3x 2 + x 1 x 3 = 2 (c) x 3/5 1 2x 2 + x 3 = 4 (d) πx 1 2x 2 + 1 3 x 3 = 7 1/3 Find the augmented matrix for each of the following systems of linear equations. (a) x 1 + 2x 2 x 4 + x 5 = 1 3x 2 + x 3 x 5 = 2 x 3 + 7x 4 = 1 (b) 4/3x 1 + x 3 x 5 = 2 5x2 + x 4 + 3x 5 = 1 x 1 + πx 2 + 9.2x 5 =.3 x 3 + (ln 23e)x 4 = e 1/5 2
2 1.2 Gaussian Elimination Leading 1s Row-Echelon Form Reduced Row-Echelon Form Free Variables Pivot Homogeneous System Trivial Solution Nontrivial Solutions Gaussian Elimination Gauss-Jordan Elimination The first nonzero number in the row that is a 1. All leading 1s have zeros below them. All leading 1s have zeros above and below them. The variables that correspond to columns without leading 1s. The variable that corresponds to a leading 1. A system of linear equations that has all zeros as their solutions. The solution when variables in a homogeneous system equal 0. Any solution to a homogeneous system such that the variables do not all equal 0. Use elementary row operations to change a matrix into Row-Echelon Form which uses back substitution to find the solution. Use elementary row operations to change a matrix into Reduced Row-Echelon Form and uses the pivots to find the solution. Solve each of the following systems using Gaussian Elimination and then Gauss-Jordan Elimination. (a) x 1 + x 2 + 2x 3 = 8 x 1 2x 2 + +3x 3 = 1 3x 1 7x 2 + 4x 3 = 10 (b) 2b + 3c = 1 3a + 6b 3c = 2 6a + 6b + 3c = 5 3
(c) 10y 4z + w = 1 x + 4y z + w = 2 3x + 2y + z + 2w = 5 2x 8y + 2z 2w = 4 x 6y + 3z = 1 (d) 2x + 2y + 4z = 0 w y 3z = 0 2w + 3x + y + z = 0 2w + x + 3y 2z = 0 Is the solution trivial? (e) x 1 + 3x 2 + x 4 = 0 x 1 + 4x 2 + 2x 3 = 0 2x 2 2x 3 x 4 = 0 2x 1 4x 2 + x 3 + x 4 = 0 x 1 2x 2 x 3 + x 4 = 0 Is the solution trivial? Find coefficients a, b, c, and d such that the following points: (0,10) (1,7) (3,-11) (4,-14) fit on the following curve: y = ax 3 + bx 2 + cx + d. 4
3 1.3 Matrices and Matrix Operations Transpose of a Matrix A = {a ij }, A T = {a ji } Trace of a Matrix trace(a) = Σa ii i Given the following matrices 3 0 4 1 1 4 2 A = 1 2 B = C = 0 2 3 1 5 1 1 1 5 2 6 1 3 D = 1 0 1 E = 1 1 2 3 2 4 4 1 3 Compute the following (a) D + E (b) 5A (c) 3(D + 2E) (d) tr(d 3E) (e) 4tr(7B) (f) 2B C (g) D T E T (h) B T + 5C T (i) (2E T 3D T ) T (j) The first row of AB (k) The third column of CA 5
4 1.4 Inverses; Rules of Matrix Arithmetic Invertible Matrix A s.t. A 1 A = I Singular Matrix A 1 s.t. A 1 A = I Inverse of a 2x2 Matrix If A = a b A 1 1 = d c d ad bc c b a Properties of the Transpose ((A) T ) T = A (A + B) T = A T + B T and (A B) T = A T B T (ka) T = ka T, where k is any scalar (AB) T = B T A T (A T ) 1 = (A 1 ) T Compute the inverses of the following matrices. 3 1 (a) 5 2 (b) 2 0 0 3 Use the given information to find A. 2 1 (a) A 1 = 3 5 (b) (I + 2A) 1 = Let A be the matrix 1 2 4 5 3 1 2 1 and p(x) = x 3 2x + 4, find p(a). If A is a square matrix and n is a positive integer, is it true that (A n ) T = (A T ) n? Justify your answer. 6
5 1.5 Elementary Matrices and a Method for Finding A 1 Elementary Matrix A matrix that represents a single row operation. Find a row operation that will restore the given elementary matrix to an identity matrix. (a) (b) (c) 1 0 0 0 1 0 0 0 3 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 7 0 0 1 0 0 0 0 1 0 0 0 0 1 Find the inverse matrix of the following matrices. 1 0 0 0 1 3 0 0 (a) 1 3 5 0 1 3 5 7 1 0 0 0 1 3 0 0 (b) 1 3 5 0 1 3 5 7 0 0 2 0 1 0 0 1 (c) 0 1 3 0 2 1 5 3 7
Consider the matrix A = such that E 2 E 1 A = I Write the matrix A = 1 0 5 2 3 2 3 1 Find elementary matrices E 1 and E 2 as a product of elementary matrices. Find the inverse of the matrix a b c d By row-reducing A I to I A 1 (Taken from Dr. Glasgow s Winter 2009 Exam 1) 8
6 1.6 Further Results on Systems of Equations and Invertibility Theorem If A is an invertible matrix the solution to A x = b is x = A 1 b. Theorem If AB is invertible, then A and B must also be invertible. Equivalent Statements If A is an nxn matrix, then the following are equivalent: (a)a is invertible (b)a x = 0 has only the trivial solution. (c)the reduced row-echelon form of A is I n (d)a is expressible as a product of elementary matrices. (e)a x = b is consistent for every nx1 matrix b (f)a x = b has exactly one solution for every nx1 matrix b Solve the system of equations using the first theorem. (a) x + y + z = 5 x + y 4z = 10 4x + y + z = 10 (b) x 1 + 2x 2 + 3x 3 = b 1 2x 1 + 5x 2 + 5x 3 = b 2 3x 1 + 5x 2 + 8x 3 = b 3 Solve the three systems of equations simultaneously. x 1 + 3x 2 + 5x 3 = b 1 x 1 2x 2 = b 2 6x 1 + 4x 2 8x 3 = b 3 (a) b 1 = 1, b 2 = 0, b 3 = 0 (b) b 1 = 0, b 2 = 1, b 3 = 1 (b) b 1 = 1, b 2 = 1, b 3 = 0 Find conditions for the b s that make the system below consistent. 9
x 1 2x 2 + 5x 3 = b 1 4x 1 5x 2 + 8x 3 = b 2 3x 1 + 3x 2 3x 3 = b 3 Suppose the system of equations A x = b has one and only one solution xɛr n for each and every bɛr n. (Evidently A is a nxn matrix.) So, tell me whether the following statement is true or is false: It is possible that the matrix A allows there to be x 0 such that A x = 0 i.e. It is possible that the homogeneous version of the above equation has a nontrivial solution. (Question taken from Dr. Glasgow s Winter 2009 Exam 1) Assuming A and B are invertible matrices of the same size, prove that (AB) 1 = B 1 A 1 (Taken from Dr. Glasgow s Winter 2009 Exam 1) Assume that both the matrix B and the matrix C are inverses of the matrix A. Show that B and C are just two aliases for the same matrix, i.e. show that in fact B = C. (Taken from Dr. Glasgow s Winter 2009 Exam 1) 10
7 1.7 Diagonal, Triangular, and Symmetric Matrices Properties of Symmetric Matricies If A and B are nxn symmetric matrices, then (a) A T is symmetric (b) A + B and A B are symmetric (c) ka is symmetric Theorem If A is an invertible symmetric matrix, then A 1 is symmetric. Theorem If A is an invertible matrix, then AA T and A T A are also invertible Compute the product by inspection. 3 0 0 0 1 0 0 0 2 2 1 4 1 2 5 Find all values of a, b, and c for which A is symmetric. 2 a 2b + 2c 2a + b + c 3 5 a + c 0 2 7 Prove that if A T A = A, then A is symmetric and A = A 2 Let A = {a ij } be an nxn matrix. a ij = i 2 + j 2 Let A = {a ij } be an nxn matrix. a ij = 2i 2 + 2j 3 Determine whether A is symmetric. Determine whether A is symmetric. 11