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Matrices $C$ and $D$ shown below are equal.

$C=\left[\begin{array}{cccc}

4 & -3 & 8 & 9.2 \\

1.2 & -6 & 5 & 1 \\

6 & 0 & -7 & 23

\end{array}\right] \quad D=\left[\begin{array}{llll}

d_{11} & d_{12} & d_{13} & d_{14}^{\prime} \\

d_{21} & d_{22} & d_{23} & d_{24} \\

d_{31} & d_{32} & d_{33} & d_{34}^{\prime}

\end{array}\right]$



What is the value of $d_{33}$?


A-7
B-6
C5
D8

Verified Answer :

  • Recognize that for two matrices to be equal, their corresponding entries must be equal.
  • Locate the element at the 3rd row and 3rd column of matrix \(C\) as \(-7\).
  • Conclude that \(d_{33}\) must be \(-7\) because \(d_{33} = c_{33}\).
  • The final value is \(\boxed{-7}\).

Explanation

  1. Understanding Matrix Equality

To determine the value of \(d_{33}\), we must rely on the definition of matrix equality. Two matrices are considered equal if and only if they have the same dimensions and all their corresponding elements are identical. Given that matrix \(C\) and matrix \(D\) are equal (\(C = D\)), every element in matrix \(D\) must be equal to the corresponding element in matrix \(C\). Specifically, the element in the \(i\)-th row and \(j\)-th column of \(D\), denoted as \(d_{ij}\), must equal the element in the \(i\)-th row and \(j\)-th column of \(C\), denoted as \(c_{ij}\).

  1. Extracting the Value of \(d_{33}\)

We are asked to find the value of \(d_{33}\). By the definition of matrix equality, \(d_{33}\) is equal to the element in the 3rd row and the 3rd column of matrix \(C\), which we denote as \(c_{33}\). Examining matrix \(C\):
\[C = \begin{bmatrix} 4 & -3 & 8 & 9.2 \ 1.2 & -6 & 5 & 1 \ 6 & 0 & -7 & 23 \end{bmatrix}\]
Locating the element in the 3rd row and 3rd column:

  • The first row is \([4, -3, 8, 9.2]\).
  • The second row is \([1.2, -6, 5, 1]\).
  • The third row is \([6, 0, -7, 23]\).

The 3rd element in the 3rd row is \(-7\). Therefore, \(c_{33} = -7\).

  1. Conclusion

Since matrix \(C\) is equal to matrix \(D\), we equate their corresponding elements. We have established that \(d_{33} = c_{33}\). Given that \(c_{33} = -7\), it follows that \(d_{33} = -7\). This matches the first option provided in the problem description. Thus, the value of \(d_{33}\) is \(\boxed{-7}\).

Examples

Matrix equality is used in various fields like computer graphics and data analysis. For instance, when two different software programs process the same dataset into a matrix format to perform calculations (like rotating an image or adjusting color levels), they verify if their resulting matrices are equal to ensure the transformation was applied consistently. Checking specific elements like \(d_{33}\) allows engineers to verify that a particular pixel or data point was calculated correctly across both systems.

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