Glossary

9.7 Solving Systems with Inverses

2 min read•june 25, 2024

Matrix inverses are powerful tools for solving systems of linear equations. They allow us to flip equations around, making it easier to find solutions. Understanding how to calculate and use these inverses is key to tackling complex problems.

Inverse matrices have real-world applications in fields like economics and engineering. By mastering these concepts, you'll be able to solve intricate systems of equations and interpret the results in meaningful ways. This skill is crucial for many advanced math and science topics.

Matrix Inverses and Systems of Linear Equations

Calculation of matrix inverses

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  • denoted as A−1A^{-1} satisfies the property A⋅A−1=A−1⋅A=IA \cdot A^{-1} = A^{-1} \cdot A = I ()
  • Finding the inverse of a 2x2 matrix [abcd]\begin{bmatrix}a & b \\ c & d\end{bmatrix}:
    • calculated as [det(A)](https://www.fiveableKeyTerm:det(A))=ad−bc[det(A)](https://www.fiveableKeyTerm:det(A)) = ad - bc
    • Inverse exists if det(A)≠0det(A) \neq 0 and given by A−1=1ad−bc[d−b−ca]A^{-1} = \frac{1}{ad-bc}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}
    • Swap positions of aa and dd, change signs of bb and cc
  • Inverting larger matrices:
    • Create augmented matrix [A∣I][A | I] (II is identity matrix same size as AA)
    • Perform transforming left side into identity matrix
    • Right side of augmented matrix becomes inverse of AA

Application of inverse matrices

  • Systems of linear equations represented in matrix form: Ax⃗=b⃗A \vec{x} = \vec{b} ()
    • AA is , x⃗\vec{x} is , b⃗\vec{b} is
  • Solving systems using :
    1. Multiply both sides by A−1A^{-1}: A−1Ax⃗=A−1b⃗A^{-1}A \vec{x} = A^{-1}\vec{b}
    2. Simplify: Ix⃗=A−1b⃗I \vec{x} = A^{-1}\vec{b}
    3. Solution given by: x⃗=A−1b⃗\vec{x} = A^{-1}\vec{b}
  • Useful for solving multiple systems with same coefficient matrix AA
    • Calculate A−1A^{-1} once, solve different systems by changing b⃗\vec{b} (supply and demand, mixture problems)
  • is used to compute the solution

Interpretation of inverse matrix solutions

  • Real-world problems often modeled by systems of linear equations ()
  • Interpreting variables:
    • Each variable represents specific quantity or attribute in problem context
  • Interpreting solutions:
    • Check if solutions make sense considering problem constraints and limitations
  • Communicating results:
    • Explain meaning of solutions in terms of original problem
    • Use appropriate units and labels when presenting results

Matrix Properties and Alternative Methods

  • : A square matrix that does not have an inverse (determinant is zero)
  • : A square matrix that has an inverse (determinant is non-zero)
  • : Another term for a nonsingular matrix
  • : An alternative method for solving systems of linear equations using determinants

Key Terms to Review (17)

A^(-1): A^(-1) represents the inverse of a matrix A. The inverse of a matrix is a unique matrix that, when multiplied by the original matrix, results in the identity matrix. The inverse of a matrix is denoted by A^(-1) and is used to solve systems of linear equations by transforming the system into an equivalent one that is easier to solve.
Coefficient Matrix: The coefficient matrix, also known as the system matrix, is a fundamental concept in linear algebra that represents the coefficients of the variables in a system of linear equations. It plays a crucial role in the analysis and solution of such systems, as well as in various applications of matrices and linear transformations.
Constant Vector: A constant vector is a vector whose components do not change, meaning its magnitude and direction remain fixed. It is a fundamental concept in linear algebra and is crucial in solving systems of linear equations using inverse matrices.
Cramer's Rule: Cramer's rule is a method used to solve systems of linear equations by expressing the solution as a ratio of determinants. It provides a systematic way to find the unique solution to a system of linear equations with the same number of variables and equations.
Det(A): The determinant of a square matrix A, denoted as det(A) or |A|, is a scalar value that provides important information about the matrix, such as whether it is invertible and how it transforms the space. The determinant is a fundamental concept in linear algebra that has numerous applications in mathematics, physics, and other fields.
Determinant: The determinant of a square matrix is a scalar value that is a function of the entries of the matrix. It carries important information about the matrix, such as whether the matrix is invertible and the volume of the parallelepiped spanned by the column vectors of the matrix.
Identity Matrix: The identity matrix is a special square matrix where all the elements on the main diagonal are 1, and all other elements are 0. It is denoted by the symbol $\mathbf{I}$ and serves as the multiplicative identity for matrix multiplication, similar to how the number 1 is the multiplicative identity for scalar multiplication.
Inverse Matrix Method: The inverse matrix method is a technique used to solve systems of linear equations by finding the inverse of the coefficient matrix and applying it to the constant terms. It provides a systematic approach to determine the unique solution to a system of linear equations, if it exists.
Invertible Matrix: An invertible matrix is a square matrix that has an inverse matrix. In other words, it is a matrix that can be multiplied by another matrix to produce the identity matrix, which means that the original matrix can be 'undone' or reversed.
Linear System: A linear system is a collection of linear equations that describe a relationship between multiple variables. It is a fundamental concept in linear algebra and is widely used in various fields, including mathematics, physics, engineering, and economics.
Matrix Inverse: The matrix inverse is a mathematical operation that allows for the solution of systems of linear equations. It is the inverse of a matrix, meaning it undoes the original matrix operation, just as division is the inverse of multiplication.
Matrix Multiplication: Matrix multiplication is a mathematical operation that combines two matrices to produce a new matrix. It is a fundamental concept in linear algebra and is essential for solving systems of linear equations and various applications in science, engineering, and data analysis.
Network Flow Problems: Network flow problems are a class of optimization problems that involve the movement of a commodity, such as goods, information, or resources, through a network of interconnected nodes and edges. These problems focus on maximizing the flow of the commodity through the network while adhering to certain constraints, such as capacity limits on the edges or the demand at the nodes.
Nonsingular Matrix: A nonsingular matrix is a square matrix that has an inverse. In other words, a nonsingular matrix is a matrix that can be inverted, meaning there exists a unique matrix that, when multiplied with the original matrix, results in the identity matrix.
Row Operations: Row operations refer to the fundamental mathematical transformations that can be performed on the rows of a matrix to solve systems of linear equations. These operations allow for the manipulation of the matrix to obtain a reduced row echelon form, which is essential for finding the solutions to the system.
Singular Matrix: A singular matrix is a square matrix that does not have an inverse. In other words, it is a matrix that cannot be multiplied by another matrix to produce the identity matrix. This means that the determinant of a singular matrix is zero, and the matrix cannot be used to solve systems of linear equations using the inverse matrix method.
Variable Vector: A variable vector is a mathematical construct that represents a quantity with both magnitude and direction, where the values of the components can change. It is a fundamental concept in linear algebra and is essential for understanding and solving systems of linear equations.
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