Finding Matrix X: Step-by-Step Solution

The problem asks us to find a matrix X based on the given equation \(2A + B + X = 0\), where matrices \(A=\begin{bmatrix} -1 & 2 \\ 3 & 4 \end{bmatrix}\) and \(B =\begin{bmatrix} 3 & -2 \\ 1 & 5 \end{bmatrix}\) are provided.

Rearranging the Matrix Equation

To find matrix X, we first rearrange the given equation: \(2A + B + X = 0\) Subtracting \(2A\) and \(B\) from both sides, we get: \(X = - (2A + B)\)

Calculating 2A

First, we need to perform scalar multiplication on matrix A. We multiply each element of matrix A by 2: $$ 2A = 2 \times \begin{bmatrix} -1 & 2 \\ 3 & 4 \end{bmatrix} $$ $$ 2A = \begin{bmatrix} 2 \times (-1) & 2 \times 2 \\ 2 \times 3 & 2 \times 4 \end{bmatrix} $$ $$ 2A = \begin{bmatrix} -2 & 4 \\ 6 & 8 \end{bmatrix} $$

Calculating 2A + B

Next, we add the resulting matrix \(2A\) to matrix B. Matrix addition involves adding corresponding elements: $$ 2A + B = \begin{bmatrix} -2 & 4 \\ 6 & 8 \end{bmatrix} + \begin{bmatrix} 3 & -2 \\ 1 & 5 \end{bmatrix} $$ $$ 2A + B = \begin{bmatrix} (-2 + 3) & (4 + (-2)) \\ (6 + 1) & (8 + 5) \end{bmatrix} $$ $$ 2A + B = \begin{bmatrix} 1 & 2 \\ 7 & 13 \end{bmatrix} $$

Calculating Matrix X

Finally, we find matrix X by taking the negative of the result from the previous step \((2A + B)\). This means multiplying each element of \((2A + B)\) by -1: $$ X = - (2A + B) $$ $$ X = - \begin{bmatrix} 1 & 2 \\ 7 & 13 \end{bmatrix} $$ $$ X = \begin{bmatrix} -1 \times 1 & -1 \times 2 \\ -1 \times 7 & -1 \times 13 \end{bmatrix} $$ $$ X = \begin{bmatrix} -1 & -2 \\ -7 & -13 \end{bmatrix} $$

Final Result Verification

The calculated matrix X is \(\begin{bmatrix} -1 & -2 \\ -7 & -13 \end{bmatrix}\). This matches one of the provided options.