9.4 Matrices 573 To explain matrix multiplication, let’s use matrices A and B that follow. A B 3 2 5 7 and 0 6 4 1 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Since matrix A contains two rows and matrix B contains two columns, the product matrix will contain two rows and two columns. To multiply two matrices, we use a row–column scheme of multiplying. The numbers in the first row of matrix A are multiplied by the numbers in the first column of matrix B. These products are then added to determine the entry in the product matrix. A B 3 2 5 7 0 6 4 1 × = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ First row First column 3 2 5 7 0 6 4 1 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ × + × = + = (3 0) (2 4) 0 8 8 The 8 is placed in the first-row, first-column position of the product matrix, . A B × The other numbers in the product matrix are obtained similarly, as illustrated in the matrix that follows. First row First column First row Second column 3 2 5 7 0 6 4 1 3 2 5 7 0 6 4 1 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ × + × = (3 0) (2 4) 8 × + × = (3 6) (2 1) 20 A B 8 20 28 37 × = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Product Matrix Second row First column Second row Second column 3 2 5 7 0 6 4 1 3 2 5 7 0 6 4 1 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ × + × = (5 0) (7 4) 28 × + × = (5 6) (7 1) 37 b) × × A B 2 2 2 3 Same Because matrix A has two columns and matrix B has two rows, the two matrices can be multiplied. The product matrix, AB, is a 2 3 × matrix. c) × × A B 2 3 2 3 Not same Because matrix A has three columns and matrix B has two rows, the two matrices cannot be multiplied. 7 Now try Exercise 33 Did You Know? The Prisoner’s Dilemma When two parties pursue conflicting interests, the situation can sometimes be described and modeled in a matrix under a branch of mathematics known as game theory . Consider a famous problem called “the prisoner’s dilemma.” A pair of suspects, A and B, are being held in separate jail cells and cannot communicate with each other. Each one is told that there are four possible outcomes: If both confess, each receives a 3-year sentence. If A confesses and B does not, A receives a 1-year sentence, whereas B receives a 10-year sentence. If B confesses and A does not, B receives a 1-year sentence and A receives a 10-year sentence. Finally, if neither confesses, each will be imprisoned for 2 years. (Try arranging this situation in a matrix.) If neither suspect knows whether the other will confess, what should each suspect do? A study of game theory shows that it is in each suspect’s best interest to confess to the crime. Matrix Multiplication
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