9.4 Matrices 571 Subtraction of Matrices Only matrices with the same dimension may be subtracted. To do so, we subtract each entry in one matrix from the corresponding entry in the other matrix. Virginia MB RB North Carolina MB RB Jan.–June July–Dec. Determine the total number of each type of bicycle sold by the corporation during each time period. Solution To solve the problem, we add matrices A and B. 515 520 425 350 290 180 250 271 1035 775 470 521 + + + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Now try Exercise 11 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = A B 515 425 290 250 520 350 180 271 Jan.–June July–Dec. MB RB MB RB We can see from the sum matrix that during the period from January through June, a total of 1035 mountain bicycles and 775 racing bicycles were sold. During the period from July through December, a total of 470 mountain bicycles and 521 racing bicycles were sold. 7 Example 3 Subtracting Matrices Determine A B − if A B 3 6 5 1 and 2 4 8 3 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Solution A B 3 6 5 1 2 4 8 3 3 2 6 ( 4) 5 8 1 ( 3) 1 10 3 2 − = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − − − − − − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 7 Now try Exercise 13 MATHEMATICS TODAY Spreadsheets One of the most useful applications of computer software is the spreadsheet, such as Excel. On a computer screen, it looks very much like an accountant’s ledger, but there the similarities end. A spreadsheet uses a matrix that assigns an identifying number and letter, in maplike fashion, to each “cell” in the matrix. The data may be dates, numbers, or sums of money. You instruct the computer what operation to perform in each column and row. You can change an entry to see what effect that has on the rest of the spreadsheet. The computer will automatically recompute all affected cells. Why This Is Important A computer has the capacity to store information in thousands of cells and can reformat the data in graph form quickly. Organizing data into a spreadsheet allows you to analyze and visualize data sets. This visualization can help in decision making and future planning. Example 4 Multiplying a Matrix by a Scalar For matrices A and B, determine (a) A3 and (b) A B 3 2 . − = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ A B 1 4 3 5 , 1 3 5 6 Solution a) The number 3 is a scalar, since we are multiplying matrix A by 3. A3 3 1 4 3 5 3(1) 3(4) 3( 3) 3(5) 3 12 9 15 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Multiplying a Matrix by a Real Number A matrix may be multiplied by a real number by multiplying each entry in the matrix by the real number. When we multiply a matrix by a real number, we call that real number a scalar . Peter Sobolev/Shutterstock
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