Perform the indicated operations.$6\begin{bmatrix}1&3&2\\-2&4&-1\end{bmatrix}+\begin{bmatrix}5&7&-1\\6&-4&7\end{bmatrix}=$6[1 3 2−2 4 −1]+[5 7 −16 −4 7]=​

1. Break Down the Problem

We need to perform the operations on a matrix expression that involves scalar multiplication and matrix addition:

1. Multiply the first matrix by the scalar 6.
2. Add the result to the second matrix.

2. Relevant Concepts

• Scalar multiplication of a matrix involves multiplying each element of the matrix by that scalar.
• Matrix addition requires both matrices to have the same dimensions, and we simply add the corresponding elements.

3. Analysis and Detail

Step 3.1: Matrix Scalar Multiplication

Given matrix: $A = \begin{bmatrix}1 & 3 & 2 \\ -2 & 4 & -1\end{bmatrix}$ Multiply by scalar 6: $6A = 6 \cdot \begin{bmatrix}1 & 3 & 2 \\ -2 & 4 & -1\end{bmatrix} = \begin{bmatrix}6 \cdot 1 & 6 \cdot 3 & 6 \cdot 2 \\ 6 \cdot -2 & 6 \cdot 4 & 6 \cdot -1\end{bmatrix}$ Calculating the elements gives: $6A = \begin{bmatrix}6 & 18 & 12 \\ -12 & 24 & -6\end{bmatrix}$

Step 3.2: Matrix Addition

Now consider the second matrix: $B = \begin{bmatrix}5 & 7 & -1 \\ 6 & -4 & 7\end{bmatrix}$ We now add the matrices $6A$ and $B$: $6A + B = \begin{bmatrix}6 & 18 & 12 \\ -12 & 24 & -6\end{bmatrix} + \begin{bmatrix}5 & 7 & -1 \\ 6 & -4 & 7\end{bmatrix}$ Performing the addition element by element: $\begin{bmatrix}6 + 5 & 18 + 7 & 12 - 1 \\ -12 + 6 & 24 - 4 & -6 + 7\end{bmatrix} = \begin{bmatrix}11 & 25 & 11 \\ -6 & 20 & 1\end{bmatrix}$

4. Verify and Summarize

Upon verifying, the calculations were correct. Therefore, we summarize our final result.

Final Answer

$6\begin{bmatrix}1 & 3 & 2 \\ -2 & 4 & -1\end{bmatrix} + \begin{bmatrix}5 & 7 & -1 \\ 6 & -4 & 7\end{bmatrix} = \begin{bmatrix}11 & 25 & 11 \\ -6 & 20 & 1\end{bmatrix}$