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Chapter 4: Problem 16

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

Short Answer

Expert verified
The given matrix does not have an inverse.

Step by step solution

01

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.
02

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.
03

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: \( ext{det}(A) = (-3)(4) - (-2)(6) = -12 + 12 = 0 \).
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Find the inverse of each matrix, if it exists. \(\left[\begin{array}{ll}{9} & {5} \\ {7} & {4}\end{array}\right]\)

Use a matrix equation to solve each system of equations. \(p-2 q=1\) \(p+5 q=22\)

Use Cramer鈥檚 Rule to solve each system of equations. \(6 x+7 y=10\) \(3 x-4 y=20\)

Solve each equation. $$ \frac{3+n}{6}=-4 $$

In Cramer鈥檚 Rule, if the value of the determinant is zero, what must be true of the graph of the system of equations represented by the determinant? Give examples to support your answer.
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