Solve for X using the matrices below.2A + 4B = −2XA = 6 −5 7 03 −4 and B = 0 7 8 0−4 −1.X =

1. Break Down the Problem

We need to solve the equation $2A + 4B = -2X$ for $X$. Given matrices are: $A = \begin{pmatrix} 6 & -5 & 7 \\ 0 & 3 & -4 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 0 & 7 & 8 \\ 0 & -4 & -1 \end{pmatrix}$

2. Relevant Concepts

1. Matrix addition.
2. Scalar multiplication of matrices.
3. Rearrangement of the equation to solve for $X$.

3. Analysis and Detail

First, calculate $2A$ and $4B$:

• Calculate $2A$: $2A = 2 \cdot \begin{pmatrix} 6 & -5 & 7 \\ 0 & 3 & -4 \end{pmatrix} = \begin{pmatrix} 12 & -10 & 14 \\ 0 & 6 & -8 \end{pmatrix}$

• Calculate $4B$: $4B = 4 \cdot \begin{pmatrix} 0 & 7 & 8 \\ 0 & -4 & -1 \end{pmatrix} = \begin{pmatrix} 0 & 28 & 32 \\ 0 & -16 & -4 \end{pmatrix}$

• Combine $2A + 4B$: $2A + 4B = \begin{pmatrix} 12 & -10 & 14 \\ 0 & 6 & -8 \end{pmatrix} + \begin{pmatrix} 0 & 28 & 32 \\ 0 & -16 & -4 \end{pmatrix} = \begin{pmatrix} 12 & 18 & 46 \\ 0 & -10 & -12 \end{pmatrix}$

Now we have: $2A + 4B = \begin{pmatrix} 12 & 18 & 46 \\ 0 & -10 & -12 \end{pmatrix}$

3. Solve for $X$

From the equation $2A + 4B = -2X$, we have: $-2X = \begin{pmatrix} 12 & 18 & 46 \\ 0 & -10 & -12 \end{pmatrix}$ This implies: $X = -\frac{1}{2} \begin{pmatrix} 12 & 18 & 46 \\ 0 & -10 & -12 \end{pmatrix} = \begin{pmatrix} -6 & -9 & -23 \\ 0 & 5 & 6 \end{pmatrix}$

4. Verify and Summarize

To verify, we substitute $X$ back into the original equation $2A + 4B = -2X$: $-2X = -2 \begin{pmatrix} -6 & -9 & -23 \\ 0 & 5 & 6 \end{pmatrix} = \begin{pmatrix} 12 & 18 & 46 \\ 0 & -10 & -12 \end{pmatrix}$ The calculations are consistent.

Final Answer

$X = \begin{pmatrix} -6 & -9 & -23 \\ 0 & 5 & 6 \end{pmatrix}$