Question

Show that each matrix has no inverse. \(\left[\begin{array}{rr}-3 & \frac{1}{2} \\ 6 & -1\end{array}\right]\)

Short answer

The determinant is zero, so the matrix has no inverse.

Step-by-step solution

Recall the condition for a matrix to have an inverse

A matrix has an inverse if and only if its determinant is non-zero. For a 2x2 matrix \[ A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \], the determinant is given by \[ \text{det}(A) = ad - bc \]

Identify the elements of the matrix

Given the matrix \[ \begin{pmatrix} -3 & \frac{1}{2} \ 6 & -1 \end{pmatrix} \], identify the elements: \[ a = -3, \ b = \frac{1}{2}, \ c = 6, \ d = -1 \]

Calculate the determinant

Using the formula for the determinant of a 2x2 matrix, calculate \[ \text{det}(A) = (-3)(-1) - \left(\frac{1}{2}\right)(6) \]. Simplify to get \[ \text{det}(A) = 3 - 3 = 0 \]

Conclude whether the matrix has an inverse

Since the determinant of the matrix is zero, \[ \text{det}(A) = 0 \], the matrix does not have an inverse.

Key concepts

Determinant Calculation

Determinant Calculation: The determinant of a matrix is a crucial value in matrix algebra. It's a scalar value that can tell us if a matrix is invertible or not.
In a more compact form, the determinant of a 2x2 matrix can be seen as:
For a given matrix \[ A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \],
the determinant is calculated using the formula:
\[ \text{det}(A) = ad - bc \]The logic behind this calculation is rooted in linear transformations. The determinant essentially represents the scaling factor by which areas (in 2D) or volumes (in higher dimensions) are altered by the transformation represented by the matrix.
If the determinant is zero, the transformation squashes the space into a lower dimension, meaning that the matrix does not have an inverse.

2x2 Matrix

Understanding 2x2 Matrix: A 2x2 matrix is one of the simplest forms of matrices and is represented as a square array with two rows and two columns.
Each element in the matrix has a specific role. For example, in the matrix \[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \], 'a', 'b', 'c', and 'd' are the elements of the matrix.
The matrix can also be interpreted as a linear transformation in a two-dimensional space, having significant applications in geometry and computer graphics.
When working with matrices, always remember to pay attention to the positions of each element, as they are key to understanding various matrix operations including addition, subtraction, and multiplication.

Matrix Algebra

Introduction to Matrix Algebra: Matrix algebra deals with the study and manipulation of matrices. It forms the backbone of many areas in mathematics and applied sciences, including computer science, engineering, and economics.
Key operations in matrix algebra include:

• Matrix Addition: Combining two matrices by adding their corresponding elements.
• Matrix Subtraction: Subtracting one matrix from another by subtracting their corresponding elements.
• Matrix Multiplication: Combining two matrices in a specific manner that involves taking the dot product of rows and columns.
• Determinant Calculation: As discussed earlier, the determinant offers insight into matrix properties, especially whether a matrix is invertible.
• Matrix Inversion: If a matrix has a non-zero determinant, it has an inverse matrix that, when multiplied with the original, yields the identity matrix.

Understanding these operations is fundamental for solving systems of linear equations, transforming geometric objects, and performing various other mathematical tasks.