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Chapter 9: Problem 21

Use a graphing calculator to find \(\mathbf{A}^{-1}\), if it exists. $$\mathbf{A}=\left[\begin{array}{rrrr}1 & 2 & 3 & 4 \\ 0 & 1 & 3 & -5 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & -1\end{array}\right]$$

Short Answer

Expert verified
The inverse of \(\textbf{A}\) is \(\begin{bmatrix} 1 & -2 & -1 & -10 \ 0 & 1 & 3 & -5 \ 0 & 0 & 1 & 2 \ 0 & 0 & 0 & -1 \end{bmatrix}\).

Step by step solution

01

- Enter the Matrix

First, turn on the graphing calculator. Access the matrix function (usually under a 'matrix' button or in the math menu). Input the matrix \(\textbf{A}\) as given: \(\textbf{A} = \begin{bmatrix}1 & 2 & 3 & 4 \ 0 & 1 & 3 & -5 \ 0 & 0 & 1 & -2 \ 0 & 0 & 0 & -1 \end{bmatrix} \). Make sure to enter each value in its respective position.
02

- Select the Inverse Function

Once Matrix \(\textbf{A}\) is entered correctly into the calculator, find and select the inverse function. This is often denoted as \( \textbf{A}^{-1} \) in the matrix menu or under a specific key related to matrix operations.
03

- Compute the Inverse

After selecting the inverse function, ensure that the matrix \(\textbf{A}\) is highlighted or selected, and press 'Enter' or the equivalent button on the calculator to compute \(\textbf{A}^{-1}\). The calculator will display the inverse matrix if it exists.
04

- Verify the Existence

Check the output. If the calculator provides a matrix, then \(\textbf{A}^{-1}\) exists. If the calculator shows an error message indicating that the matrix is singular or does not have an inverse, then \(\textbf{A}^{-1}\) does not exist.
05

- Write Down the Inverse

If an inverse exists, copy the resulting inverse matrix from the calculator display. For \(\textbf{A}\), the inverse \(\textbf{A}^{-1}\) is: \(\textbf{A}^{-1} = \begin{bmatrix} 1 & -2 & -1 & -10 \ 0 & 1 & 3 & -5 \ 0 & 0 & 1 & 2 \ 0 & 0 & 0 & -1 \end{bmatrix} \).

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Key Concepts

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

inverse matrices
An inverse matrix, often denoted as \(\textbf{A}^{-1}\), is a matrix that, when multiplied by the original matrix \(\textbf{A}\), results in an identity matrix \(\textbf{I}\). Not all matrices have an inverse; those that do are called invertible or non-singular. Matrices that don't have an inverse are referred to as singular. For a matrix to be invertible, it must be square (i.e., having the same number of rows and columns) and have a non-zero determinant. The relationship is \(\textbf{A} \times \textbf{A}^{-1} = \textbf{I}\). To find the inverse of a matrix, you can use methods like Gauss-Jordan elimination, the adjoint method, or a graphing calculator to simplify the process.
graphing calculator
A graphing calculator is a handheld device capable of plotting graphs, solving simultaneous equations, and performing many other mathematical computations, including matrix operations. To find the inverse of a matrix using a graphing calculator, follow these steps:
  • Access the matrix menu, typically under a 'matrix' button or within a math function menu.
  • Enter your matrix by specifying each element in the correct position.
  • Select the inverse function (often labeled as \(\textbf{A}^{-1}\) in the matrix menu).
  • Execute the operation to compute the inverse.
If the inverse exists, the calculator will display it. If not, you will get an error indicating that the matrix is singular.
linear algebra
Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces but is particularly concerned with the properties of vectors and matrices. Core topics in linear algebra include
  • Vector and matrix operations
  • Determinants and inverses
  • Eigenvalues and eigenvectors
  • Systems of linear equations
Linear algebra has various applications in multiple fields such as physics, computer science, engineering, and economics. Understanding matrix operations and inverses are fundamental skills needed to solve complex problems in these areas.
matrix operations
Matrix operations are fundamental procedures used in linear algebra. These operations include addition, subtraction, scalar multiplication, and multiplication of matrices. For example:
  • **Addition and Subtraction**: Matrices of the same dimensions can be added or subtracted element-wise.
  • **Scalar Multiplication**: Each element in the matrix is multiplied by a scalar (a single number).
  • **Matrix Multiplication**: The dot product of rows and columns forms a new matrix. This operation is not commutative, meaning \( \textbf{A} \times \textbf{B} eq \textbf{B} \times \textbf{A} \).
Advanced matrix operations include finding determinants, computing inverses, and eigenvalues. You can use these operations to solve systems of equations, perform transformations, and much more in various scientific and engineering fields.

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