Answered

What is the value of $D$ in the matrix equation below?



$D+\left[\begin{array}{cc}

-1 & -15 \\

0 & 7

\end{array}\right]=\left[\begin{array}{cc}

8 & -4 \\

-14 & 12

\end{array}\right]$


A$\left[\begin{array}{cc}8 & 11 \\ 14 & 5\end{array}\right]$


B$\left[\begin{array}{cc}9 & -19 \\ 14 & 5\end{array}\right]$


C$\left[\begin{array}{cc}9 & -19 \\ -14 & 5\end{array}\right]$


D$\left[\begin{array}{cc}9 & 11 \\ -14 & 5\end{array}\right]$

Verified Answer :

  • Isolate matrix \(D\) by subtracting \(\begin{bmatrix} -1 & -15 \ 0 & 7 \end{bmatrix}\) from both sides of the equation.
  • Perform the subtraction element-wise for each component.
  • Calculate the values: \(8 - (-1) = 9\), \(-4 - (-15) = 11\), \(-14 - 0 = -14\), and \(12 - 7 = 5\).
  • Identify the final matrix: \(\boxed{\begin{bmatrix} 9 & 11 \ -14 & 5 \end{bmatrix}}\).

Explanation

  1. Problem Analysis

The problem asks to solve for matrix \(D\) in the equation \(D + \begin{bmatrix} -1 & -15 \ 0 & 7 \end{bmatrix} = \begin{bmatrix} 8 & -4 \ -14 & 12 \end{bmatrix}\). Matrix addition and subtraction are performed element-wise, meaning corresponding components are added or subtracted independently. To find \(D\), we need to isolate it on one side of the equation.

  1. Isolating Matrix D

To isolate \(D\), we subtract the matrix \(\begin{bmatrix} -1 & -15 \ 0 & 7 \end{bmatrix}\) from both sides of the equation:
\[D = \begin{bmatrix} 8 & -4 \ -14 & 12 \end{bmatrix} - \begin{bmatrix} -1 & -15 \ 0 & 7 \end{bmatrix}\]
This subtraction is performed by subtracting each corresponding element of the second matrix from the first.

  1. Calculating Components of D

We perform the subtraction element by element:

  • For the element in the first row and first column: \(8 - (-1) = 8 + 1 = 9\).
  • For the element in the first row and second column: \(-4 - (-15) = -4 + 15 = 11\).
  • For the element in the second row and first column: \(-14 - 0 = -14\).
  • For the element in the second row and second column: \(12 - 7 = 5\).

Thus, the resulting matrix is:
\[D = \begin{bmatrix} 9 & 11 \ -14 & 5 \end{bmatrix}\]

  1. Conclusion

Comparing the calculated matrix \(D = \begin{bmatrix} 9 & 11 \ -14 & 5 \end{bmatrix}\) with the given options, we identify that the result matches the fourth option. The value of matrix \(D\) is \(\boxed{\begin{bmatrix} 9 & 11 \ -14 & 5 \end{bmatrix}}\).

Examples

Matrix operations are essential in various fields, such as computer graphics and data analysis. For instance, if you are transforming a 2D image represented by coordinate matrices, you might need to add or subtract translation matrices to shift the position of objects. Solving equations like these helps in finding the necessary transformation parameters to achieve a specific shift or movement of elements in a grid-based system.

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