1. **Problem statement:** Given real numbers $a$ and $b$, determine the value of the matrix or expression $ \begin{bmatrix} 2x - 1 \\ x - 2 \end{bmatrix} $. 2. **Understanding the problem:** The problem seems to ask for the evaluation or simplification of the vector or matrix with components $2x - 1$ and $x - 2$ where $x$ is a variable. 3. **No specific value for $x$ is given, so we express the vector in terms of $x$:** $$\mathbf{v} = \begin{bmatrix} 2x - 1 \\ x - 2 \end{bmatrix}$$ 4. **If the problem is to find the magnitude (norm) of this vector, we use the formula:** $$\|\mathbf{v}\| = \sqrt{(2x - 1)^2 + (x - 2)^2}$$ 5. **Expanding the squares:** $$ (2x - 1)^2 = 4x^2 - 4x + 1 $$ $$ (x - 2)^2 = x^2 - 4x + 4 $$ 6. **Sum the two:** $$ 4x^2 - 4x + 1 + x^2 - 4x + 4 = 5x^2 - 8x + 5 $$ 7. **Therefore, the magnitude is:** $$ \|\mathbf{v}\| = \sqrt{5x^2 - 8x + 5} $$ 8. **This is the simplified form of the magnitude of the vector for any real $x$.** **Final answer:** $$ \boxed{\begin{bmatrix} 2x - 1 \\ x - 2 \end{bmatrix} \text{ and } \|\mathbf{v}\| = \sqrt{5x^2 - 8x + 5}} $$