Problem 6 - Precalculus: Mathematical for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{l} 3 \\ 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is not square, so its determinant does not exist.

Step 1: Understanding the Matrix

The given matrix is a column matrix with two rows:\[\begin{bmatrix}3 \0 \\end{bmatrix}\]It is a 2x1 matrix, meaning there are 2 rows and 1 column.

Step 2: Determine Matrix Type

To find a determinant, the matrix must be square (same number of rows and columns). Since the given matrix is 2x1, it is not square.

Step 3: Confirm the Determinant Existence

Determinants exist only for square matrices. Since our matrix is not square, we cannot calculate a determinant for it.

Key Concepts

Square Matrix

A square matrix is an essential concept when dealing with determinants. It is defined as a matrix that has the same number of rows as it does columns. For instance, a 2x2 matrix or a 3x3 matrix are both square matrices because they have equal dimensions.

Square matrices are unique because they allow you to compute the determinant, a special number that can provide insights into the matrix's properties. The determinant can tell us about the invertibility of the matrix, among other characteristics.

• If a matrix has a determinant of zero, it is not invertible (also known as singular).
• If the determinant is non-zero, the matrix is invertible (non-singular).

Square matrices are not limited by the number of rows or columns, as long as these numbers are equal. They play a crucial role in linear algebra and systems of equations.

Non-Square Matrix

Non-square matrices have different numbers of rows and columns, such as a 2x3 matrix (2 rows, 3 columns) or a 5x2 matrix (5 rows, 2 columns). Unlike their square counterparts, non-square matrices do not have determinants. This absence is because determinants require an equal number of rows and columns to compute.

The inability to determine a determinant doesn't diminish the utility of non-square matrices. They are prevalent in various applications, like transforming coordinates, where the number of inputs doesn't match the number of outputs. Since non-square matrices do not meet the condition for calculating determinants, other mathematical tools like matrix multiplication or transpose are used to analyze them.

Matrix Rows and Columns

Understanding the basic structure of matrices begins with rows and columns, which form the building blocks of these arrays of numbers. The number of rows runs horizontally across the matrix, while the number of columns runs vertically. These dimensions are usually described as 'm by n,' where 'm' represents the number of rows and 'n' represents the number of columns.

For example, in the case of our original matrix \[\begin{bmatrix}3 \0 \end{bmatrix}\]this is a 2x1 matrix. Here, 2 represents the number of rows, and 1 denotes the single column present.

• Rows often represent scenarios or data points, like in a dataset.
• Columns usually represent attributes or variables.

Understanding the arrangement of rows and columns is foundational for interpreting matrices and their respective operations.