Problem 15 - Algebra 2

Question

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{ll}{2} & {3} \\ {1} & {1}\end{array}\right] $$

Step-by-Step Solution

Answer

The given matrix has an inverse, which is \[\left[\begin{array}{cc}-1 & 3\1 & -2\end{array}\right]\].

Step 1: Determine the Determinant

Calculate the determinant of the matrix. The determinant of a 2x2 matrix represented as \[\left[\begin{array}{cc}a & b\c & d\end{array}\right]\] is \(ad - bc\). For our matrix, \(ad - bc = (2)(1) - (3)(1) = 2 - 3 = -1\).

Step 2: Check if the Matrix has an Inverse

A matrix has an inverse if and only if its determinant is non-zero. Since the determinant is -1, which is not equal to zero, the matrix does have an inverse.

Step 3: Find the Inverse Matrix

The inverse of a 2x2 matrix is given by \[\frac{1}{ad - bc}\left[\begin{array}{cc}d & -b\-c & a\end{array}\right]\]. Applying this formula, the inverse of our matrix is \[\frac{1}{-1}\left[\begin{array}{cc}1 & -3\-1 & 2\end{array}\right]\], which simplifies to \[\left[\begin{array}{cc}-1 & 3\1 & -2\end{array}\right]\].

Key Concepts

Determinant of a Matrix

Understanding the determinant of a matrix is fundamental when dealing with matrix operations, especially when determining whether a given matrix can be inverted. The determinant can be thought of as a scalar value that encodes certain properties of a matrix. For a 2x2 matrix, which is represented as
\[ \left[\begin{array}{cc}a & b\ c & d\end{array}\right] \],
the determinant is calculated using the formula: \( ad - bc \). This operation gives us a single number that helps us decide if a matrix is invertible. A non-zero determinant indicates that the matrix does indeed have an inverse, while a zero determinant means there is no inverse, which signals that the matrix is 'singular', or not invertible.
The significance of the determinant goes beyond just finding the inverse; it has geometrical interpretations and is used in various areas including systems of linear equations, eigenvalues, and eigenvectors.

2x2 Matrix Inversion

Inverting a 2x2 matrix is a handy skill, particularly when dealing with systems of equations or transformations in linear algebra. The formula to find the inverse of a 2x2 matrix is quite straightforward:
\[ \frac{1}{ad - bc}\left[\begin{array}{cc}d & -b\ -c & a\end{array}\right] \],
where \( ad - bc \) is the determinant. Each element of the original matrix has a specific role in this formula. The positions of the elements \(a\) and \(d\) are swapped, while the signs of the elements \(b\) and \(c\) are changed. It's crucial to remember that this formula only works if the determinant is non-zero, as division by zero is not defined. The process of inverting matrices is vital in various computational applications, where matrices often represent transformations and the inverse corresponds to reversing those transformations.

Non-zero Determinant

The importance of a non-zero determinant can't be overstated when discussing matrix invertibility. A non-zero determinant indicates that a matrix has a unique inverse and that the system it represents has a single solution. This concept is central to many areas of algebra and calculus. When the determinant of a matrix is zero, it signals that the system may have no solutions or an infinite number of solutions, which corresponds to a case where the matrix represents a transformation that squashes a higher-dimensional space into a lower one.
In the context of 2x2 matrices, a non-zero determinant verifies that our square matrix doesn't reduce the plane to a line or a single point, allowing for an inverse operation which essentially 'undoes' the transformation encoded by the matrix.

Matrix Operations

Matrix operations form the foundation of linear algebra and play an integral role in various mathematical applications. These operations include addition, subtraction, multiplication, finding the determinant, and matrix inversion. Each operation has specific rules and properties that must be followed.
For instance, matrix multiplication is not commutative, meaning that the order in which matrices are multiplied matters. Also, the process of obtaining the inverse of a matrix is itself an operation that requires adherence to certain conditions, such as the existence of a non-zero determinant. Understanding matrix operations enables students to solve more complex problems, like finding eigenvalues, diagonalizing matrices, and applying linear transformations in mathematical modeling, physics, computer graphics, and more.