Problem 16 - Algebra 2

Question

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rr}{-3} & {-2} \\ {6} & {4}\end{array}\right] $$

Step-by-Step Solution

Answer

The given matrix does not have an inverse.

Step 1: Understand Matrix Inversion

The inverse of a matrix \( A \) is denoted as \( A^{-1} \) and is defined such that \( A \times A^{-1} = I \), where \( I \) is the identity matrix. Not all matrices have inverses; a matrix must be square and have a non-zero determinant to have an inverse.

Step 2: Identify the Matrix and Check Dimensions

The given matrix is \( A = \begin{bmatrix} -3 & -2 \ 6 & 4 \end{bmatrix} \). This is a 2x2 square matrix, which means it is eligible for having an inverse, provided its determinant is non-zero.

Step 3: Calculate the Determinant

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ext{det}(A) = ad - bc \). Substitute the given values: \( ext{det}(A) = (-3)(4) - (-2)(6) = -12 + 12 = 0 \).

Step 4: Determine Invertibility

Since the determinant of the matrix is zero (\( ext{det}(A) = 0 \)), the matrix is singular, which means it does not have an inverse.

Key Concepts

Matrix Inversion

Matrix inversion is a crucial concept in linear algebra. It involves finding a matrix that "undoes" the operation of the original matrix. If a matrix is denoted as \( A \), its inverse is represented as \( A^{-1} \). When you multiply a matrix by its inverse, it results in the identity matrix, \( I \), such that \( A \times A^{-1} = I \). Not all matrices can be inverted. Only matrices that are square (having the same number of rows as columns) and have a non-zero determinant can have an inverse. If a matrix has an inverse, it is known as a non-singular or invertible matrix. If no inverse exists, the matrix is singular.

Determinant

The determinant is a special number calculated from a square matrix. It provides important information about the matrix, such as whether an inverse exists. For a 2x2 matrix, the determinant \( \text{det}(A) \) is calculated as \( ad - bc \) for a matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \).In the exercise, the determinant was calculated for the matrix \( \begin{bmatrix} -3 & -2 \ 6 & 4 \end{bmatrix} \) and yielded \( 0 \). This indicates that the matrix is singular and therefore does not have an inverse. The determinant provides a quick way to assess the invertibility of a matrix. It's a non-invertible scenario when it's exactly zero.

Square Matrix

A square matrix is defined as a matrix with the same number of rows and columns. Examples include 2x2, 3x3, or \( n \times n \) matrices. This shape is essential in the context of matrix inversion because only square matrices can potentially have inverses.In our example, the given matrix \( \begin{bmatrix} -3 & -2 \ 6 & 4 \end{bmatrix} \) is a 2x2 matrix. Before attempting to find the inverse or calculating the determinant, one must first confirm that the matrix is square. This is a basic but fundamental requirement for matrix operations like inversion.

Identity Matrix

An identity matrix acts as the multiplicative "one" in matrix algebra. It is symbolized as \( I \) and features 1s along its diagonal from top left to bottom right, with all other elements being 0. For a 2x2 identity matrix, it looks like \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).The role of an identity matrix in the context of inversion is pivotal. If \( A \) is a matrix with an inverse \( A^{-1} \), multiplying \( A \) by \( A^{-1} \), or vice versa, yields the identity matrix: \( A \times A^{-1} = I \). It ensures that the matrix has been "+undone," paralleling the need for \( 1 \) in regular multiplication for invertible results.