<p>For what values of \(x\): \([1\quad 2\quad 1]\begin{bmatrix}1 & 2 & 0\\2 & 0 & 1\\1 & 0 & 2\end{bmatrix}\begin{bmatrix}0\\2\\x\end{bmatrix} = 0\,?</p>
Step-by-Step Solution
Key Concept: Multiply the matrices sequentially from left to right: first the row vector by the square matrix, then the result by the column vector. The final product must equal zero to find x.
<p><strong>Step 1:</strong> Multiply the row vector [1 2 1] by the 3×3 matrix:</p><p>[1 2 1]<begin>bmatrix}1 & 2 & 0\2 & 0 & 1\1 & 0 & 2\end{bmatrix} = [1(1)+2(2)+1(1), 1(2)+2(0)+1(0), 1(0)+2(1)+1(2)]</p><p>= [1+4+1, 2+0+0, 0+2+2] = [6, 2, 4]</p><p><strong>Step 2:</strong> Multiply the result [6 2 4] by the column vector [0, 2, x]<sup>T</sup>:</p><p>[6 2 4]·[0, 2, x]<sup>T</sup> = 6(0) + 2(2) + 4(x) = 0 + 4 + 4x</p><p><strong>Step 3:</strong> Set equal to zero:</p><p>4 + 4x = 0</p><p>4x = -4</p><p>x = -1</p><p>∴ Answer: x = -1</p>
Correct Answer: -1