Matrix A below gives the fraction of the earth's area and the projected fraction of its population for five continents in 2050. Matrix B gives the earth's area (in square miles) and its projected 2050 population.† Fraction of Area Fraction of Population 0.162 0.047 North America 0.119 0.086 South America A = 0.066 0.065 Europe 0.298 0.582 Asia 0.202 0.215 Africa Area Population B = 57,850,000 0 Area 0 9,322,000,000 Population Find the area (in square miles) and population of each given continent by finding AB.

In order to get a 5×2 matrix result from the multiplication of AB, the dimensions of matrix B must be 2×2. The given numbers can be put on the diagonal of that matrix to achieve the desired effect.

You want ...

[tex]A=\left[\begin{array}{cc}.162&.047\\.119&.086\\.066&.065\\.298&.582\\.202&.215\end{array}\right],\ B=\left[\begin{array}{cc}57.850\cdot 10^6&0\\0&9.322\cdot 10^9\end{array}\right] \\\\AB\approx\left[\begin{array}{cc}9.37\cdot 10^6&.438\cdot 10^9\\6.88\cdot 10^6&.802\cdot 10^9\\3.82\cdot 10^6&.606\cdot 10^9\\17.2\cdot 10^6&5.43\cdot 10^9\\11.7\cdot 10^6&2.00\cdot 10^9\end{array}\right][/tex]

The rows are in the order given in the problem statement: North America, South America, Europe, Asia, Africa. The left column is area; the right column is population. Here, numbers have been rounded to 3 significant figures. The multipliers are consistently millions and billions to make comparison easier.