Problem 17 - College Algebra with Corequisite Support

Question

For the following exercises, find the multiplicative inverse of each matrix, if it exists. $$ \left[\begin{array}{ll} 1 & 1 \\ 2 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix has no inverse because its determinant is zero.

Step 1: Check matrix dimensions

The given matrix \( \begin{bmatrix} 1 & 1 \ 2 & 2 \end{bmatrix} \) is a 2x2 matrix. Inverses exist only for square matrices, which this is, so we can proceed to the next step.

Step 2: Calculate the determinant

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( ad - bc \). For our matrix, \( a = 1, b = 1, c = 2, \) and \( d = 2 \). Thus, the determinant is:\[1 \times 2 - 1 \times 2 = 0\]

Step 3: Verify invertibility

A matrix is invertible only if its determinant is non-zero. Since the determinant of our matrix is 0, this matrix is not invertible.

Key Concepts

Determinant

The determinant is a special number that can tell us a lot about a matrix. Specifically, for 2x2 matrices, it helps to determine if the matrix has an inverse or not.
The determinant of a 2x2 matrix of the form \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\] is calculated using the formula \( ad - bc \).

• "a" and "d" are the diagonal elements.
• "b" and "c" are the off-diagonal elements.

This simple multiplication and subtraction can indicate matrix properties like invertibility.
If this number equals zero, the matrix is singular and does not have an inverse. If it is non-zero, the matrix is invertible and thus an inverse can be found.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a square matrix. It has exactly two rows and two columns, forming a compact square shape. This type of matrix is often used in linear algebra due to its simplified operations and calculations.
Here's an example of a basic 2x2 matrix: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]These matrices are handy for various computations, including finding determinants, performing matrix operations, and solving systems of linear equations.
With matrix operations so fundamental in areas like computer graphics, physics, and engineering, understanding how to work with smaller matrices is a great way to build up your skills for handling larger, more complex matrices.

Matrix Invertibility

Matrix invertibility is a crucial concept in linear algebra. When a matrix is invertible, it means that there exists another matrix, called the inverse, which when multiplied with the original matrix, results in the identity matrix.

• The identity matrix for a 2x2 matrix is \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).

However, not every matrix is invertible. The key factor that determines invertibility is the determinant.
For a matrix to be invertible, its determinant should not be zero. A zero determinant means that the matrix is singular, and thus, it does not have an inverse. This property is particularly important when solving systems of linear equations, as a non-invertible matrix implies that unique solutions may not exist for the system.