4.4 Invertible Matrices and Their Properties

Invertible matrices are the superheroes of linear algebra. They're square matrices with special powers – they have multiplicative inverses. This means you can undo their transformations, making them crucial for solving equations and understanding linear systems.

Knowing if a matrix is invertible is key. You can check by finding its determinant – if it's not zero, you're good to go. Invertible matrices have unique inverses and follow special rules for multiplication, transposition, and determinants. They're essential for solving linear equations.

Invertible Matrices

Definition and Notation

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• An invertible matrix, also known as a non-singular matrix, is a square matrix that has a multiplicative inverse
• The inverse of a matrix A is denoted as $A^{-1}$, and when multiplied together, the result is the identity matrix: $AA^{-1} = A^{-1}A = I$
• A matrix is invertible if and only if its determinant is non-zero
• The identity matrix is always invertible, and it is its own inverse ($I^{-1} = I$)

Determining Invertibility

• To determine if a matrix is invertible, calculate its determinant
  • If the determinant is non-zero, the matrix is invertible
  • If the determinant is zero, the matrix is not invertible (singular)
• For 2x2 matrices, the determinant is calculated using the formula: $det(A) = ad - bc$, where a, b, c, and d are the elements of the matrix
• For larger matrices, the determinant can be calculated using various methods such as cofactor expansion, row reduction, or Laplace expansion
• A matrix is not invertible if its rows or columns are linearly dependent

Invertibility of Matrices

Calculating Determinants

• The determinant of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is given by $det(A) = ad - bc$
  • Example: For the matrix $A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$, $det(A) = 2 \cdot 5 - 3 \cdot 4 = -2$
• For larger matrices, the determinant can be calculated using cofactor expansion
  • The cofactor of an element $a_{ij}$ is $C_{ij} = (-1)^{i+j} \cdot M_{ij}$, where $M_{ij}$ is the minor of $a_{ij}$ (the determinant of the submatrix formed by deleting the i-th row and j-th column)
  • The determinant is the sum of the products of elements in any row or column and their respective cofactors: $det(A) = \sum_{j=1}^n a_{ij} \cdot C_{ij}$ for any fixed i
• Laplace expansion and row reduction can also be used to calculate determinants for larger matrices

Linear Dependence and Invertibility

• A matrix is not invertible if its rows or columns are linearly dependent
  • Linearly dependent rows or columns indicate that the matrix has a determinant of zero
• If a matrix has a row or column of zeros, or if two rows or columns are scalar multiples of each other, the matrix is not invertible
  • Example: The matrix $A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$ is not invertible because the second row is a scalar multiple of the first row

Matrix Inversion by Row Reduction

Augmented Matrix Method

• To find the inverse of a matrix using row reduction, create an augmented matrix by placing the identity matrix to the right of the original matrix
  • For an n x n matrix A, the augmented matrix is $[A | I_n]$, where $I_n$ is the n x n identity matrix
• Perform row operations (row switching, row scaling, and row addition) on the augmented matrix until the left side becomes the identity matrix
  • The goal is to transform the augmented matrix into $[I_n | A^{-1}]$
• The right side of the augmented matrix will then be the inverse of the original matrix
  • Example: To find the inverse of $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, create the augmented matrix $[A | I_2] = \begin{bmatrix} 1 & 2 & 1 & 0 \\ 3 & 4 & 0 & 1 \end{bmatrix}$ and perform row operations until the left side becomes $I_2$

Non-Invertible Matrices

• If the left side cannot be reduced to the identity matrix, the original matrix is not invertible
  • This occurs when a row of zeros appears on the left side during the row reduction process
• In such cases, the matrix has no inverse, and the system of linear equations it represents either has no solution or infinitely many solutions
  • Example: The matrix $A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$ is not invertible, as row reduction will lead to a row of zeros on the left side of the augmented matrix

Properties of Invertible Matrices

Uniqueness and Multiplication

• The inverse of an invertible matrix is unique
  • If A is invertible and $AB = I$ and $CA = I$, then $B = C = A^{-1}$
• If A and B are invertible matrices of the same size, then $(AB)^{-1} = B^{-1}A^{-1}$
  • The inverse of a product of invertible matrices is equal to the product of their inverses in reverse order
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, then $(AB)^{-1} = B^{-1}A^{-1}$

Transpose and Determinant

• The transpose of the inverse of a matrix A is equal to the inverse of the transpose of A: $(A^{-1})^T = (A^T)^{-1}$
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, then $(A^{-1})^T = (A^T)^{-1}$
• If A is an invertible matrix, then $A^{-1}$ is also invertible, and $(A^{-1})^{-1} = A$
• The determinant of the inverse of a matrix A is the reciprocal of the determinant of A: $det(A^{-1}) = 1/det(A)$
  • Example: If $det(A) = 5$, then $det(A^{-1}) = 1/5$

Matrix Inversion for Linear Systems

Solving Systems of Linear Equations

• Matrix inversion can be used to solve systems of linear equations in the form $Ax = b$, where A is the coefficient matrix, x is the vector of variables, and b is the constant vector
• If A is invertible, the solution to the system can be found by multiplying both sides of the equation by $A^{-1}$: $A^{-1}Ax = A^{-1}b$, which simplifies to $x = A^{-1}b$
  • Example: To solve the system $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}x = \begin{bmatrix} 5 \\ 11 \end{bmatrix}$, find $A^{-1}$ and multiply it by b to obtain x
• This method is particularly useful when solving multiple systems of linear equations with the same coefficient matrix A but different constant vectors b

Existence and Uniqueness of Solutions

• When using matrix inversion to solve systems of linear equations, it is important to ensure that the coefficient matrix A is invertible; otherwise, the system may have no solution or infinitely many solutions
• If A is invertible, the system $Ax = b$ has a unique solution for any constant vector b
• If A is not invertible, the system may have no solution (if b is not in the column space of A) or infinitely many solutions (if b is in the column space of A)
  • Example: The system $\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}x = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ has infinitely many solutions because the coefficient matrix is not invertible