808 CHAPTER 11 Systems of Equations and Inequalities Historical Feature Matrices were invented in 1857 by Arthur Cayley (1821—1895) as a way of efficiently computing the result of substituting one linear system into another (see Historical Problem 3). The resulting system had incredible richness, in the sense that a wide variety of mathematical systems could be mimicked by the matrices. Cayley and his friend James J. Sylvester (1814—1897) spent much of the rest of their lives elaborating the theory. The torch was then passed to Georg Frobenius (1849—1917), whose deep investigations established a central place for matrices in modern mathematics. In 1924, rather to the surprise of physicists, it was found that matrices (with complex numbers in them) were exactly the right tool for describing the behavior of atomic systems. Today, matrices are used in a wide variety of applications. Arthur Cayley (1821–1895) 1. Matrices and Complex Numbers Frobenius emphasized in his research how matrices could be used to mimic other mathematical systems. Here, we mimic the behavior of complex numbers using matrices. Mathematicians call such a relationship an isomorphism . ↔ Complex number Matrix ↔ + − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ i a b a b b a Note that the complex number can be read off the top line of the matrix. Then ↔ ↔ + − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − i i and 2 3 2 3 3 2 4 2 2 4 4 2 (a) Find the matrices corresponding to − i 2 5 and + i 1 3 . (b) Multiply the two matrices. (c) Find the corresponding complex number for the matrix found in part (b). (d) Multiply − i 2 5 and + i 1 3. The result should be the same as that found in part (c). The process also works for addition and subtraction. Try it for yourself. 2. Compute ( )( ) + − a bi a bi using matrices. Interpret the result. 3. Cayley’s Definition of Matrix Multiplication Cayley devised matrix multiplication to simplify the following problem: = + = + ⎧ ⎨ ⎪⎪ ⎩⎪⎪ = + = + ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ u ar bs v cr ds x ku lv y mu nv (a) Find x and y in terms of r and s by substituting u and v from the first system of equations into the second system of equations. (b) Use the result of part (a) to find the 2 by 2 matrix A in ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ x y A r s (c) Now look at the following way to do it. Write the equations in matrix form. ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ u v a b c d r s x y k l m n u v So ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ x y k l m n a b c d r s Do you see how Cayley defined matrix multiplication? Historical Problems 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Concepts and Vocabulary 11.4 Assess Your Understanding 1. A matrix that has the same number of rows as columns is called a(n) matrix. 2. True or False Matrix addition is commutative. 3. True or False If A and B are square matrices, then AB BA. = 4. Suppose that A is a square n by n matrix that is nonsingular. The matrix B for which AB BA In = = is the of the matrix A. 5. True or False The identity matrix has properties similar to those of the real number 1. 6. If AX B = represents a matrix equation where A is a nonsingular matrix, then we can solve the equation using X = . 7. Multiple Choice To find the product AB of two matrices A and B, which statement must be true? (a) The number of columns in A must equal the number of rows in B. (b) The number of rows in A must equal the number of columns in B. (c) A and B must have the same number of rows and the same number of columns. (d) A and B must both be square matrices. 8. Multiple Choice A matrix that has no inverse is called a n : ( ) (a) zero matrix (b) nonsingular matrix (c) identity matrix (d) singular matrix Credit: Library of Congress
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