State The Dimensions Of The Matrix.$\[ A=\left[\begin{array}{ll} -9 & 1 \\ -7 & 5 \\ -5 & 8 \end{array}\right] \\]A. \[$3 \times 3\$\] B. \[$2 \times 2\$\] C. \[$2 \times 3\$\] D. \[$3 \times 2\$\]
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The dimensions of a matrix are essential in understanding its properties and operations. In this article, we will explore how to determine the dimensions of a matrix, using a given example.
Understanding Matrix Dimensions
The dimensions of a matrix are denoted by the number of rows and columns it contains. For instance, a matrix with 3 rows and 2 columns is denoted as a 3x2 matrix. The number of rows is always specified first, followed by the number of columns.
Example Matrix
Let's consider the following matrix:
To determine the dimensions of this matrix, we need to count the number of rows and columns.
Counting Rows and Columns
Looking at the matrix, we can see that it has 3 rows:
- The first row contains the elements -9 and 1.
- The second row contains the elements -7 and 5.
- The third row contains the elements -5 and 8.
Now, let's count the number of columns. We can see that each row has 2 elements, which means the matrix has 2 columns.
Determining the Dimensions
Based on our count, we can conclude that the matrix A has 3 rows and 2 columns. Therefore, the dimensions of the matrix A are 3x2.
Conclusion
In conclusion, determining the dimensions of a matrix is a straightforward process that involves counting the number of rows and columns. By understanding the dimensions of a matrix, we can perform various operations, such as addition, subtraction, multiplication, and division.
Answer
The correct answer is:
D. ${3 \times 2\$}
Additional Examples
To reinforce our understanding, let's consider a few more examples:
- A matrix with 2 rows and 3 columns is denoted as a 2x3 matrix.
- A matrix with 4 rows and 1 column is denoted as a 4x1 matrix.
- A matrix with 1 row and 5 columns is denoted as a 1x5 matrix.
Tips and Tricks
When determining the dimensions of a matrix, make sure to count the number of rows and columns carefully. A common mistake is to confuse the number of rows and columns.
Common Mistakes
- Counting the number of rows incorrectly.
- Counting the number of columns incorrectly.
- Confusing the number of rows and columns.
Conclusion
In conclusion, determining the dimensions of a matrix is a fundamental concept in mathematics. By understanding the dimensions of a matrix, we can perform various operations and solve problems in mathematics. Remember to count the number of rows and columns carefully to avoid common mistakes.
Final Answer
The final answer is:
In the previous article, we discussed how to determine the dimensions of a matrix. In this article, we will answer some frequently asked questions about matrix dimensions.
Q: What are the dimensions of a matrix with 5 rows and 2 columns?
A: The dimensions of a matrix with 5 rows and 2 columns are 5x2.
Q: How do I determine the dimensions of a matrix with 3 rows and 4 columns?
A: To determine the dimensions of a matrix with 3 rows and 4 columns, simply count the number of rows and columns. In this case, the matrix has 3 rows and 4 columns, so the dimensions are 3x4.
Q: What are the dimensions of a matrix with 1 row and 6 columns?
A: The dimensions of a matrix with 1 row and 6 columns are 1x6.
Q: Can a matrix have 0 rows or 0 columns?
A: Yes, a matrix can have 0 rows or 0 columns. For example, a matrix with 0 rows and 3 columns is denoted as a 0x3 matrix.
Q: What are the dimensions of a matrix with 2 rows and 0 columns?
A: The dimensions of a matrix with 2 rows and 0 columns are 2x0. However, it's worth noting that a matrix with 0 columns is not a valid matrix, as it would not have any elements.
Q: How do I determine the dimensions of a matrix with a variable number of rows and columns?
A: To determine the dimensions of a matrix with a variable number of rows and columns, you can use variables to represent the number of rows and columns. For example, if a matrix has m rows and n columns, the dimensions are mxn.
Q: Can a matrix have a negative number of rows or columns?
A: No, a matrix cannot have a negative number of rows or columns. The number of rows and columns must be a non-negative integer.
Q: What are the dimensions of a matrix with an infinite number of rows and columns?
A: A matrix with an infinite number of rows and columns is not a valid matrix, as it would not have any finite dimensions.
Q: How do I determine the dimensions of a matrix with a fractional number of rows or columns?
A: A matrix cannot have a fractional number of rows or columns. The number of rows and columns must be a non-negative integer.
Conclusion
In conclusion, determining the dimensions of a matrix is a fundamental concept in mathematics. By understanding the dimensions of a matrix, we can perform various operations and solve problems in mathematics. Remember to count the number of rows and columns carefully to avoid common mistakes.
Final Answer
The final answer is:
- A matrix with 5 rows and 2 columns has dimensions 5x2.
- A matrix with 3 rows and 4 columns has dimensions 3x4.
- A matrix with 1 row and 6 columns has dimensions 1x6.
- A matrix with 0 rows and 3 columns has dimensions 0x3.
- A matrix with 2 rows and 0 columns has dimensions 2x0.
- A matrix with m rows and n columns has dimensions mxn.
- A matrix cannot have a negative number of rows or columns.
- A matrix cannot have a fractional number of rows or columns.
- A matrix with an infinite number of rows and columns is not a valid matrix.