If $$ A=\left|\begin{array}{ccc} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & x \end{array}\right| $$ is an idempotent matrix, then \(\mathrm{x}\) is equal to \(\begin{array}{llll}\text { (a) }-1 & \text { (b) }-5 & \text { (c) }-4 & \text { (d) }-3\end{array}\)

To find A * A, we'll multiply the matrix A by itself. This involves taking the dot product of each row of the left matrix with each column of the right matrix, and placing the result in the corresponding cell of the resulting matrix. Here's the calculation: \(A * A = \left(\begin{array}{ccc} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & x \end{array}\right) \left(\begin{array}{ccc} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & x \end{array}\right)\) \(A * A = \left(\begin{array}{ccc} (2 * 2) + (-2 * -1) + (-4 * 1) & (2 * -2)+(-2 * 3) +(-4 * -2) & (2 * -4 )+(-2 * 4) +(-4 * x) \\ (-1 * 2) + (3 * -1 ) + (4 * 1) & (-1 * -2) + (3 * 3) + (4 * -2) & (-1 * -4) + (3 * 4) + (4 * x) \\ (1 * 2) + (-2 * -1) + (x * 1) & (1 * -2) + (-2 * 3) + (x * -2) & (1 * -4) + (-2 * 4) + (x * x) \end{array}\right)\) After calculating all these dot products, we get: \(A * A = \left(\begin{array}{ccc} 6 & 0 & -12-4x \\ -1 & 13 & 0 \\ 3 & -8 & -12+x^2 \end{array}\right)\)

Since A * A = A for an idempotent matrix, we can compare the elements of the resulting matrix product to the original matrix A. Let's look at the element in the last column of the first row. In the original matrix, this element is -4. In the product matrix (A * A), this element is -12-4x. We can set up an equation: -4 = -12-4x

Now, we'll solve this equation for x: -4 = -12 - 4x Add 12 to both sides: 8 = -4x Now, divide both sides by -4: x = -2 However, notice that when we plug x back into A, A and A * A are not equal, which means that there is no solution for x that makes A an idempotent matrix and none of the given answer choices are valid. The answer choice should include the option "not possible" or "no solution."