Which Matrix Is Equal To ${ \left[\begin{array}{ccc}-6 & -6.5 & 1.7 \ 2 & -8.5 & 19.3\end{array}\right] }$?A. ${ \left[\begin{array}{cc}6 & 2 \ 6.5 & 8.5 \ 1.7 & 19.3\end{array}\right] }$B. $[ \left[\begin{array}{cc}-6 & 2
Introduction
In mathematics, matrices are a fundamental concept used to represent linear transformations and solve systems of equations. Matrix equality is a crucial aspect of linear algebra, where two matrices are considered equal if they have the same dimensions and corresponding elements are equal. In this article, we will explore the concept of matrix equality and determine which matrix is equal to the given matrix.
Understanding Matrix Equality
Matrix equality is based on the concept of equality of two matrices. Two matrices A and B are said to be equal if they have the same dimensions and corresponding elements are equal. This means that if A and B are two matrices of the same size, then:
A = B if and only if a_ij = b_ij for all i and j
where a_ij and b_ij are the elements of matrices A and B, respectively.
The Given Matrix
The given matrix is:
{ \left[\begin{array}{ccc}-6 & -6.5 & 1.7 \\ 2 & -8.5 & 19.3\end{array}\right] \}
This matrix has two rows and three columns, making it a 2x3 matrix.
Option A
Option A is:
{ \left[\begin{array}{cc}6 & 2 \\ 6.5 & 8.5 \\ 1.7 & 19.3\end{array}\right] \}
This matrix has three rows and two columns, making it a 3x2 matrix. Since the dimensions of the given matrix and Option A are different, they cannot be equal.
Option B
Option B is:
{ \left[\begin{array}{cc}-6 & 2 \\ 6.5 & 8.5 \\ 1.7 & 19.3\end{array}\right] \}
This matrix has three rows and two columns, making it a 3x2 matrix. However, the corresponding elements of the given matrix and Option B are not equal. For example, the element in the first row and first column of the given matrix is -6, while the element in the first row and first column of Option B is -6.
Option C
Option C is:
{ \left[\begin{array}{cc}-6 & -6.5 & 1.7 \\ 2 & -8.5 & 19.3\end{array}\right] \}
This matrix has two rows and three columns, making it a 2x3 matrix. The corresponding elements of the given matrix and Option C are equal, making them equal.
Conclusion
In conclusion, the matrix equal to the given matrix is Option C:
{ \left[\begin{array}{cc}-6 & -6.5 & 1.7 \\ 2 & -8.5 & 19.3\end{array}\right] \}
This matrix has the same dimensions and corresponding elements as the given matrix, making them equal.
Final Answer
The final answer is Option C.
References
- [1] Linear Algebra and Its Applications, Gilbert Strang
- [2] Matrix Algebra, James E. Gentle
- [3] Introduction to Linear Algebra, Gilbert Strang
Additional Resources
- [1] Khan Academy: Linear Algebra
- [2] MIT OpenCourseWare: Linear Algebra
- [3] Wolfram MathWorld: Matrix Algebra
Matrix Equality Q&A =====================
Introduction
In our previous article, we explored the concept of matrix equality and determined which matrix is equal to the given matrix. In this article, we will answer some frequently asked questions related to matrix equality.
Q: What is matrix equality?
A: Matrix equality is a concept in linear algebra where two matrices are considered equal if they have the same dimensions and corresponding elements are equal.
Q: How do I determine if two matrices are equal?
A: To determine if two matrices are equal, you need to check if they have the same dimensions and if the corresponding elements are equal. You can do this by comparing the elements of the two matrices, row by row and column by column.
Q: What are the conditions for two matrices to be equal?
A: Two matrices A and B are equal if and only if they have the same dimensions and corresponding elements are equal. This means that if A and B are two matrices of the same size, then:
A = B if and only if a_ij = b_ij for all i and j
where a_ij and b_ij are the elements of matrices A and B, respectively.
Q: Can two matrices with different dimensions be equal?
A: No, two matrices with different dimensions cannot be equal. For example, a 2x3 matrix and a 3x2 matrix cannot be equal, even if their corresponding elements are equal.
Q: Can two matrices with the same dimensions but different elements be equal?
A: No, two matrices with the same dimensions but different elements cannot be equal. For example, two matrices with the same dimensions but different elements in the same position cannot be equal.
Q: How do I find the equal matrix of a given matrix?
A: To find the equal matrix of a given matrix, you need to find a matrix that has the same dimensions and corresponding elements as the given matrix. You can do this by comparing the elements of the given matrix, row by row and column by column.
Q: What are some common mistakes to avoid when working with matrix equality?
A: Some common mistakes to avoid when working with matrix equality include:
- Comparing matrices with different dimensions
- Comparing matrices with the same dimensions but different elements
- Not checking if the corresponding elements are equal
- Not checking if the matrices have the same dimensions
Q: How do I apply matrix equality in real-world problems?
A: Matrix equality is used in many real-world problems, including:
- Linear transformations
- Systems of equations
- Data analysis
- Machine learning
Conclusion
In conclusion, matrix equality is a fundamental concept in linear algebra that is used to determine if two matrices are equal. By understanding the conditions for matrix equality and how to apply it in real-world problems, you can solve complex problems and make informed decisions.
Final Answer
The final answer is that matrix equality is a concept in linear algebra where two matrices are considered equal if they have the same dimensions and corresponding elements are equal.
References
- [1] Linear Algebra and Its Applications, Gilbert Strang
- [2] Matrix Algebra, James E. Gentle
- [3] Introduction to Linear Algebra, Gilbert Strang
Additional Resources
- [1] Khan Academy: Linear Algebra
- [2] MIT OpenCourseWare: Linear Algebra
- [3] Wolfram MathWorld: Matrix Algebra