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Chapter 8: Problem 12

Exer. \(3-12:\) Find the inverse of the matrix if it exists. $$\left[\begin{array}{lll} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{array}\right]$$

Short Answer

Expert verified
The matrix does not have an inverse because its determinant is zero.

Step by step solution

01

Check the Matrix Determinant

To find the inverse of a matrix, the first step is to check if the determinant is zero. If it is zero, the matrix does not have an inverse. Calculate the determinant of the matrix:\[ \text{det}(A) = \begin{vmatrix} 1 & 1 & 1 \ 2 & 2 & 2 \ 3 & 3 & 3 \end{vmatrix} \]For a 3x3 matrix, the determinant is calculated using cofactor expansion along any row or column. Let's choose the first row:\[ \text{det}(A) = 1 \times \begin{vmatrix} 2 & 2 \ 3 & 3 \end{vmatrix} - 1 \times \begin{vmatrix} 2 & 2 \ 3 & 3 \end{vmatrix} + 1 \times \begin{vmatrix} 2 & 2 \ 3 & 3 \end{vmatrix} \]Each minor is a 2x2 matrix with determinant calculated as:\[ \begin{vmatrix} 2 & 2 \ 3 & 3 \end{vmatrix} = (2 \cdot 3) - (2 \cdot 3) = 0 \]Substitute back into the cofactor expansion:\[ \text{det}(A) = 1 \times 0 - 1 \times 0 + 1 \times 0 = 0 \]
02

Conclusion based on Determinant

Since the determinant of the matrix is zero, the matrix does not have an inverse. For an inverse to exist, the determinant must be non-zero.

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Key Concepts

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

Matrix Determinant
The determinant of a matrix is a special number that can be calculated from its elements. It provides important insights into the properties of a matrix. One key use of the matrix determinant is to determine if a matrix has an inverse or not. To find the determinant of a 3x3 matrix, you typically use the method called cofactor expansion. Let's break this down into simpler terms.

Consider a matrix:
  • The determinant helps in understanding different characteristics, like whether the matrix is invertible (an important property for solving linear equations).
  • If the determinant is 0, the matrix is termed "singular," meaning it does not have an inverse.
  • Computing the determinant often involves reducing the matrix size through processes like cofactor expansion.
In simpler words, think of the determinant as a special instruction on whether you can "reverse" the matrix operations or if it's stuck without a way back.
3x3 Matrix
A 3x3 matrix is a rectangular array of numbers consisting of 3 rows and 3 columns. It's often used in various fields like computer graphics, physics, and engineering due to its ability to handle complex data.

Understanding the layout of a 3x3 matrix is crucial for operations like finding determinants and inverses. Here's a quick breakdown:
  • Each element in a 3x3 matrix is identified by its position, usually denoted as \(a_{ij}\), where \(i\) represents the row and \(j\) represents the column.
  • To perform operations, you often need to understand each position's contribution to calculations (like in cofactor expansion).
  • Working with a 3x3 matrix allows you to understand how individual elements impact the entire matrix operation outcome.
Essentially, imagining this grid with nine smaller blocks assists in making operations simpler and visualizing how transformations affect the matrix as a whole.
Cofactor Expansion
Cofactor expansion is a method used to calculate the determinant of larger matrices, like 3x3 matrices. It involves breaking down the matrix into smaller determinants, specifically using minors, which are themselves 2x2 determinants.

Here's a simple way to understand cofactor expansion:
  • Choose any row or column of a matrix you wish to expand across.
  • For each element in the selected row or column, determine its minor by omitting the row and column of that element.
  • Calculate the determinant of each minor (2x2 matrix) and multiply it by the original element, adjusting the sign based on the element's position.
Each product forms a component of the final determinant calculation. The process may seem detailed, but with practice, it simplifies the determination of whether a matrix is invertible. This method is often used in linear algebra calculations due to its systematic approach.
Non-invertible Matrix
A non-invertible matrix, also known as a singular matrix, is a matrix that does not have an inverse. The absence of an inverse is typically due to a determinant that equals zero. Here's why this happens:

  • When the determinant of a matrix is zero, it indicates the matrix cannot be inverted. This is because the inverse of a matrix, mathematically, requires dividing by the determinant.
  • A non-invertible matrix often represents systems of equations without a unique solution or matrices that compress data along certain dimensions, leading to loss of information.
  • In practical terms, trying to "reverse" matrix operations with a determinant of zero is like trying to undo a tightening knot without any slack.
Understanding why a matrix does not have an inverse is critical, especially in applications like solving linear systems, where an invertible matrix is crucial to finding solutions uniquely and efficiently.

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Most popular questions from this chapter

Verify the Identity by expanding each determinant. $$\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|+\left|\begin{array}{ll} a & e \\ c & f \end{array}\right|=\left|\begin{array}{ll} a & b+e \\ c & d+f \end{array}\right|$$

Find the partial fraction decomposition. \(\frac{x^{5}-5 x^{4}+7 x^{3}-x^{2}-4 x+12}{x^{3}-3 x^{2}}\)

Let \(I=I_{2}\) be the identity matrix of order 2, and let \(f(x)=|\boldsymbol{A}-\boldsymbol{x} \boldsymbol{I}| .\) Find (a) the polynomial \(f(\boldsymbol{x})\) and (b) the zeros of \(f(x)\). (In the study of matrices, \(f(x)\) is the characteristic polynomial of \(A,\) and the zeros of \(f(x)\) are the characteristic values (eigenvalues) of \(A .\) ) $$A=\left[\begin{array}{rr} -3 & -2 \\ 2 & 2 \end{array}\right]$$

Express the determinant in the form \(a i+b j+c k\) for real numbers \(a, b,\) and \(c\) $$\left|\begin{array}{rrr} i & j & k \\ 2 & -1 & 6 \\ -3 & 5 & 1 \end{array}\right|$$

The data in the table are generated by the function \(f .\) Graphically approximate the unknown constants \(a\) and \(b\) to four decimal places. $$f(x)=a \ln b x$$ $$\begin{array}{|cc|} \hline x & f(x) \\ \hline 1 & -8.2080 \\ 2 & -11.7400 \\ \hline 3 & -13.8061 \\ \hline 4 & -15.2720 \\ \hline \end{array}$$
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