Compute the products using (a) the definition, as in Example 1 (b) the row–vector rule for computing Ax. If a product is undefined, explain why.

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EXAMPLE 1 The elimination procedure is shown here with and without matrix nota- tion, and the results are placed side by side for comparison: x12x2 + x3 = 0 2x28x3 = 8 - 5x3 = 10 5 5x1 Keep x1 in the first equation and eliminate it from the other equations. To do so, add -5 times equation 1 to equation 3. After some practice, this type of calculation is usually performed mentally: 1-2 1 0 02-8 8 0-5 -5x₁ + 10x₂ 5x1 -5. [equation 1] 5x3 = 0 5x3 = 10 + [equation 3] [new equation 3] 10x₂10x3 = 10 The result of this calculation is written in place of the original third equation: x₁2x₂ + x3 = 0 2x₂8x₂ = 8 10x2 - 10x3 = 10 x₁2x₂ + x₂ = 0 4x3 = 4 x2 - 10x₂10x3 = 10 x₁2x₂ + x2 - Now, multiply equation 2 by in order to obtain 1 as the coefficient for x2. (This calculation will simplify the arithmetic in the next step.) 10 1 -2 1 0 8 0 2-8 0 10 -10 0 x3 = 4x3 = 4 30x3 = -30 1 -2 0 1 -4 0 10-10 Use the x₂ in equation 2 to eliminate the 10x2 in equation 3. The "mental" computation is 10 1 0 4 10 -10-[equation 2] + [equation 3] [new equation 3] The result of this calculation is written in place of the previous third equation (row): -10x2 + 40x3 = -40 10x2 - 10x3 = 10 30x3 = -30 1 0 0 -2 1 0 1 -4 4 0 30 -30