Problem 48 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the given matrix. $$ \left[\begin{array}{rrrr} -1 & 0 & 2 & 5 \\ 0 & 0 & 0 & 2 \\ 0 & 2 & -1 & 0 \\ 1 & -3 & 0 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The given matrix is not invertible because its determinant is zero.

Step 1: Check if the matrix is invertible

A matrix is invertible if it is a square matrix and its determinant is non-zero. First, we ensure this matrix is square. It is a 4x4 matrix, which satisfies the square requirement. Next, we need to calculate the determinant to check if it is invertible.

Step 2: Calculate the determinant

Use the formula for the determinant of a 4x4 matrix. The determinant is calculated through expansion by minors. After performing the relevant row and column operations, calculate the determinant of smaller matrices, then sum up these results factoring in the signs. For this matrix, the determinant is found to be zero.

Step 3: Conclude on the invertibility

Since the determinant of the matrix is zero, this matrix is not invertible. A zero determinant indicates that the matrix is singular and does not have an inverse.

Key Concepts

Determinant Calculation

The determinant is a special number calculated from a matrix that can help us understand its properties. For instance, it tells us if the matrix has an inverse. Specifically, a matrix without an inverse has a determinant of zero. For a square matrix like the one given, calculating the determinant involves several steps.
Start by using a method called expansion by minors, which is a bit like breaking down the matrix into smaller sections.

• Identify a row or a column to expand on. Typically, it’s best to choose one with the most zeros to simplify calculations.
• For each element in your chosen row or column, compute the determinant of the resulting smaller matrix, called a minor, and multiply it by the element.
• It is essential to consider the alternating signs (+ or -) starting with a positive sign for the top-left element.

Ultimately, when this is done for the given matrix, the determinant equals zero. This indicates a singular or non-invertible matrix.

Square Matrix

A square matrix is a fundamental type of matrix in linear algebra, having the same number of rows and columns. Understanding this concept is crucial because only square matrices can have determinants or be invertible.
Here are some key aspects of square matrices:

• They can be of any order, meaning a 2x2, 3x3, 4x4, etc., as long as the number of rows equals the number of columns.
• Matrix operations like finding the determinant or the inverse apply only to square matrices.
• In our example, the matrix is 4x4 so, the determinant calculation and the potential for inversion are possible, provided non-zero determinant exists.

Recognizing and working with square matrices is essential for advancing in linear algebra, as many problems and equations are based on them.

Singular Matrix

A singular matrix is one of the most crucial concepts in linear algebra, especially when dealing with inversion. It directly relates to the determinant calculation.
A matrix is singular if its determinant is zero. Here’s what this means and why it matters:

• Such a matrix does not have an inverse. In the context of solving equations, this can imply that no unique solution exists or infinitely many solutions exist.
• Singular matrices can often indicate dependencies between rows or columns, meaning they aren’t independent, which can lead to important insights in applications like systems of equations.
• Checking whether a matrix is singular is one of the first steps when attempting to find an inverse, as it saves time by avoiding complex computations on non-invertible matrices.

Understanding that a zero determinant leads to a singular matrix helps us conclude that the matrix can't be inverted.