- Matrix equality requires both identical dimensions and identical corresponding entries.
- The original matrix has dimensions \(2 \times 3\).
- Options 1 and 2 are \(3 \times 2\) matrices and do not match the dimensions.
- Option 3 has the correct dimensions but different values.
- Option 4 matches both dimensions and all entries of the original matrix, which is \(\boxed{\begin{bmatrix} -6 & -6.5 & 1.7 \ 2 & -8.5 & 19.3 \end{bmatrix}}\).
Explanation
- Understanding Matrix Equality
To determine which matrix is equal to a given matrix \(A = \begin{bmatrix} -6 & -6.5 & 1.7 \ 2 & -8.5 & 19.3 \end{bmatrix}\), we must follow the definition of matrix equality. Two matrices are equal if and only if they have the exact same dimensions (same number of rows and same number of columns) and all corresponding entries are identical.
The given matrix \(A\) is a \(2 \times 3\) matrix, meaning it has 2 rows and 3 columns.
Its entries are:
- Row 1: \(-6, -6.5, 1.7\)
- Row 2: \(2, -8.5, 19.3\)
- Evaluating the Options
We will now compare the given matrix \(A\) against each of the four provided options:
- Option 1: \(\begin{bmatrix} 6 & 2 \ 6.5 & 8.5 \ 1.7 & 19.3 \end{bmatrix}\)
This matrix has dimensions \(3 \times 2\) (3 rows and 2 columns). Since its dimensions do not match the \(2 \times 3\) dimensions of matrix \(A\), it cannot be equal.
- Option 2: \(\begin{bmatrix} -6 & 2 \ -6.5 & -8.5 \ 1.7 & 19.3 \end{bmatrix}\)
This matrix also has dimensions \(3 \times 2\). It does not match the dimensions of matrix \(A\).
- Option 3: \(\begin{bmatrix} 8 & 6.5 & -1.7 \ -2 & 8.5 & -19.3 \end{bmatrix}\)
This matrix is a \(2 \times 3\) matrix, so it has the same dimensions as matrix \(A\). However, we must check if all corresponding entries are identical:
- Row 1, Column 1: \(8
eq -6\)
- Row 1, Column 2: \(6.5
eq -6.5\)
- ...and so on. Since the entries are not the same, this matrix is not equal to \(A\).
- Option 4: \(\begin{bmatrix} -6 & -6.5 & 1.7 \ 2 & -8.5 & 19.3 \end{bmatrix}\)
This is a \(2 \times 3\) matrix. Comparing its entries to matrix \(A\):
- Row 1: \(-6 = -6\), \(-6.5 = -6.5\), \(1.7 = 1.7\)
- Row 2: \(2 = 2\), \(-8.5 = -8.5\), \(19.3 = 19.3\)
All corresponding entries are identical. Therefore, this matrix is equal to matrix \(A\).
- Conclusion
Based on the comparison of dimensions and entries, we have determined that the fourth option is the only matrix equal to the original matrix \(A\).
The final answer is \(\boxed{\begin{bmatrix} -6 & -6.5 & 1.7 \ 2 & -8.5 & 19.3 \end{bmatrix}}\).
Examples
In data management, matrix equality is fundamental when comparing datasets represented in table form. For example, if two different computer systems store sensor readings from a vehicle in a specific row-column format, verifying that the matrices are equal ensures that both systems have received and processed the identical set of data without errors. If the matrices were not identical, it would indicate a corruption or mismatch in the data stream.