\( \left[ \begin{array} { c c } { x - 2 } & { 4 - x } \\ { 3 } & { x } \end{array} \right] = 0 \)

To solve the matrix equation \( \left[ \begin{array} { c c } { x - 2 } & { 4 - x } \\ { 3 } & { x } \end{array} \right] = 0 \), we need to set each element of the matrix to zero. This gives us the following equations: 1. \( x - 2 = 0 \) 2. \( 4 - x = 0 \) 3. \( 3 = 0 \) 4. \( x = 0 \) Looking at these equations, we can quickly identify that the third equation, \( 3 = 0 \), is not valid, meaning there is no solution for this matrix equation that satisfies all elements being zero. The inconsistencies in the equations suggest that the matrix does not represent a valid system that can be solved; hence, the system is inconsistent. This little exercise provides a peek into matrix theory where it’s vital to check not just the variables, but also the logical coherence of the equations derived from them! In real-world applications, matrices are used in everything from computer graphics transformations to solving systems of equations in engineering. For example, in robotics, matrices help to determine the position and movement of parts in a 3D space, making them essential for designing and controlling robots effectively.