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What is the sum of the matrices shown below?



$\left[\begin{array}{ccc}

4 & 19 & -5 \\

7 & 0 & -14

\end{array}\right] + \left[\begin{array}{ccc}

-8 & 7 & 0 \\

-1 & 17 & 6

\end{array}\right]=$


A$\left[\begin{array}{ccc}

11 & 19 & 19 \\

8 & 24 & 8

\end{array}\right]$


B$\left[\begin{array}{ccc}

11 & 19 & -19 \\

-9 & 24 & 8

\end{array}\right]$


C$\left[\begin{array}{ccc}

-4 & 26 & 5 \\

8 & 17 & 8

\end{array}\right]$


D$\left[\begin{array}{ccc}

-4 & 26 & -5 \\

6 & 17 & -8

\end{array}\right]$

Verified Answer :

  • Identify the matrices \(A\) and \(B\) of the same dimensions (2x3).
  • Apply the component-wise addition rule: add corresponding elements at each row and column position.
  • Perform the additions: \(4-8=-4\), \(19+7=26\), \(-5+0=-5\), \(7-1=6\), \(0+17=17\), and \(-14+6=-8\).
  • Match the resulting matrix to the correct option: \(\boxed{\begin{bmatrix} -4 & 26 & -5 \ 6 & 17 & -8 \end{bmatrix}}\).

Explanation

  1. Problem Analysis and Matrix Addition Rule

The objective is to compute the sum of two 2x3 matrices, denoted by \(A\) and \(B\). A matrix is a rectangular array of numbers, and adding two matrices of the same dimensions is performed by adding their corresponding elements. Given:
\[A = \begin{bmatrix} 4 & 19 & -5 \ 7 & 0 & -14 \end{bmatrix}, \quad B = \begin{bmatrix} -8 & 7 & 0 \ -1 & 17 & 6 \end{bmatrix}\]
To find the sum \(C = A + B\), we calculate each element \(c_{ij}\) where \(c_{ij} = a_{ij} + b_{ij}\). The resulting matrix \(C\) will also be a 2x3 matrix.

  1. Performing the Calculation

We perform the component-wise addition for each of the 6 entries in the matrices:

  • For the first row:
  • The first entry: \(c_{11} = 4 + (-8) = -4\)
  • The second entry: \(c_{12} = 19 + 7 = 26\)
  • The third entry: \(c_{13} = (-5) + 0 = -5\)
  • For the second row:
  • The first entry: \(c_{21} = 7 + (-1) = 6\)
  • The second entry: \(c_{22} = 0 + 17 = 17\)
  • The third entry: \(c_{23} = (-14) + 6 = -8\)

Combining these values, we get:
\[C = \begin{bmatrix} -4 & 26 & -5 \ 6 & 17 & -8 \end{bmatrix}\]

  1. Final Conclusion

Comparing our calculated result \(C = \begin{bmatrix} -4 & 26 & -5 \ 6 & 17 & -8 \end{bmatrix}\) to the provided options, we can see that it matches the fourth choice exactly. Therefore, the sum of the two matrices is:
\[\boxed{\begin{bmatrix} -4 & 26 & -5 \ 6 & 17 & -8 \end{bmatrix}}\]

Examples

Matrix addition is a fundamental tool in various fields, such as computer graphics and data analysis. Imagine you are managing the inventory of two different warehouse locations, each represented by a 2x3 matrix of stock levels for three different types of products over two seasons. Adding these two matrices allows you to quickly calculate the combined inventory count for each category and season across both locations simultaneously, making logistics and resource planning much more efficient.

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