Verified Answer :

  • Identify the matrix as a rectangular array of numbers.
  • Count the horizontal rows: there are 2.
  • Count the vertical columns: there are 4.
  • State the dimensions as \(R \times C\), which is \(\boxed{2 \times 4}\).

Explanation

  1. Understanding Matrix Dimensions

The matrix provided is:
\[A = \begin{bmatrix} 3 & 1 & 8 & -1 \ 5 & 0 & -2 & 0 \end{bmatrix}\]
A matrix is a rectangular array of numbers arranged in horizontal rows and vertical columns. The dimensions of a matrix are expressed as \(R \times C\), where \(R\) represents the total number of horizontal rows and \(C\) represents the total number of vertical columns. Our objective is to identify these two values for the given matrix.

  1. Counting the Rows

To determine the number of rows (\(R\)), we count the horizontal lines of elements in the matrix:

  • First row: \(3, 1, 8, -1\)
  • Second row: \(5, 0, -2, 0\)

There are 2 horizontal rows. Thus, \(R = 2\).

  1. Counting the Columns

To determine the number of columns (\(C\)), we count the vertical lines of elements in the matrix:

  • First column: \(3, 5\)
  • Second column: \(1, 0\)
  • Third column: \(8, -2\)
  • Fourth column: \(-1, 0\)

There are 4 vertical columns. Thus, \(C = 4\).

  1. Determining the Final Dimensions

Having identified that there are 2 rows and 4 columns, the dimensions of the matrix are given by the product \(R \times C\). Therefore, the dimensions of the matrix are \(2 \times 4\). The final answer is \(\boxed{2 \times 4}\).

Examples

In data science and programming, matrices are fundamental for storing and manipulating structured data. For instance, if you have a dataset representing the grades of 2 students (rows) across 4 different subjects (columns), you would store this as a \(2 \times 4\) matrix. Understanding matrix dimensions is the first step in performing operations like matrix addition or multiplication, which are essential for machine learning algorithms and statistical analysis.

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