Which of the following describes the dimensions of matrix A?
[tex]A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{23}\end{array}\right][/tex]
A. [tex]2 \times 3[/tex]
B. [tex]6 \times 1[/tex]
C. [tex]3 \times 2[/tex]
D. [tex]3 \times 3[/tex]
Answer (1)
The dimensions of a matrix are determined by the number of rows and columns. Matrix A has 3 rows and 2 columns. The dimensions of matrix A are therefore 3 × 2 . The final answer is 3 × 2 .
Explanation
Identifying the Matrix The matrix A is given as: A = [ a 11 a 12 a 21 a 22 a 31 a 23 ] . We need to determine the dimensions of this matrix.
Understanding Matrix Dimensions To find the dimensions of a matrix, we count the number of rows and the number of columns. The number of rows is the number of horizontal lines of elements, and the number of columns is the number of vertical lines of elements.
Counting Rows In matrix A, we can see that there are 3 rows: Row 1: [ a 11 , a 12 ]
Row 2: [ a 21 , a 22 ]
Row 3: [ a 31 , a 23 ]
Counting Columns Similarly, we can see that there are 2 columns: Column 1: [ a 11 a 21 a 31 ]
Column 2: [ a 12 a 22 a 23 ]
Determining Dimensions Therefore, the dimensions of matrix A are 3 rows by 2 columns, which is written as 3 × 2 .
Final Answer Comparing our result with the given options, we find that the correct answer is 3 × 2 .
Examples
Matrix dimensions are crucial in various applications, such as image processing. For instance, if you have a grayscale image represented as a matrix, the dimensions tell you the image's height and width in pixels. A 3 × 2 matrix could represent a tiny image with 3 rows and 2 columns of pixels. Understanding matrix dimensions helps in manipulating and processing images effectively, like resizing or applying filters.
