Linear Algebra: Cramer's Rule and Matrix Inverse

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Questions and Answers

What is the eigenvalue corresponding to the eigenvector x = (-5/6)?

  • 2
  • 4
  • -2
  • -4 (correct)

If Ay = (5/6 2/6)(1/1) equals λ(1/1), what is the value of λ?

  • 4 (correct)
  • -4
  • 0
  • 2

Which of the following statements is true about eigenvectors?

  • There can be infinitely many eigenvectors corresponding to a single eigenvalue. (correct)
  • An eigenvector can never be parallel to another eigenvector.
  • Eigenvectors must always be unique.
  • Eigenvectors can only be found if the eigenvalue is real.

What condition must be met for λ to be considered an eigenvalue?

<p>The system must be homogeneous. (C)</p> Signup and view all the answers

Which of the following expressions represents the rearranged equation of Ax = λx?

<p>(A - λ)x = 0 (A)</p> Signup and view all the answers

For the matrix A with eigenvalue λ = 2, how is the eigenspace determined?

<p>By solving (A - λI)x = 0 (A)</p> Signup and view all the answers

What indicates that z = (-1/1) is not an eigenvector?

<p>Az does not equal a scaled version of z. (A)</p> Signup and view all the answers

What is the first step in applying Cramer's Rule to solve the system Ax = b?

<p>Replace column # i of A with b (A), Calculate the determinant of the matrix A (C)</p> Signup and view all the answers

Which formula represents the correct way to compute the solution for x2 using Cramer's Rule?

<p>x<sub>2</sub> = det(A<sub>2</sub>)/det(A) (C)</p> Signup and view all the answers

What property of determinants is established by Theorem 5?

<p>Determinants of a matrix and its transpose are equal (C)</p> Signup and view all the answers

How can the determinant of the product of two matrices A and B be expressed?

<p>det(AB) = det(A) * det(B) (B)</p> Signup and view all the answers

In the context of finding the inverse of matrix A, what does the cofactor Cij represent?

<p>The determinant of the matrix A without row i and column j (A)</p> Signup and view all the answers

What is a necessary condition for a matrix A to have an inverse using the determinant?

<p>det(A) must be non-zero (D)</p> Signup and view all the answers

What does an eigenvalue represent in the context of a matrix A?

<p>A scalar such that Ax = λx for some non-zero vector x (D)</p> Signup and view all the answers

What is true about the relationship between eigenvectors and their corresponding eigenvalues?

<p>Eigenvectors remain unchanged under the transformation by A (C)</p> Signup and view all the answers

What is the condition for the matrix A - λI to have eigenvectors?

<p>det(A - λI) = 0 (B)</p> Signup and view all the answers

What do the vectors (1/2) and (-3/2) represent in the context of the eigenvalue problem?

<p>They are eigenvectors corresponding to λ = 2. (A)</p> Signup and view all the answers

Which equation is equivalent to the non-trivial solution of the system (A - 2I)x = 0?

<p>2x1 - x2 + 6x3 = 0 (D)</p> Signup and view all the answers

What can be inferred about the eigenvalues of a triangular matrix?

<p>The eigenvalues are the diagonal entries themselves. (A)</p> Signup and view all the answers

If A is a (3 x 3) triangular matrix with diagonal entries a11, a22, and a33, what is the correct expression for det(A - λI)?

<p>(a11 - λ)(a22 - λ)(a33 - λ) (D)</p> Signup and view all the answers

Which of the following describes the eigenspace associated with an eigenvalue?

<p>It is the span of all possible eigenvectors for that eigenvalue. (A)</p> Signup and view all the answers

What does the term 'free variables' refer to in the context of the equation derived from (A - 2I)x = 0?

<p>Variables that can take any value without restriction. (C)</p> Signup and view all the answers

Which statement is true regarding the eigenvalues based on the singularity of the matrix A - λI?

<p>A - λI is singular if and only if λ is an eigenvalue of A. (D)</p> Signup and view all the answers

Flashcards

Cramer's Rule

A method to solve a system of linear equations (Ax = b) where A is an n x n matrix. Individual solution components are calculated by ratios of determinants.

Eigenvalue

A scalar (number) λ that scales an eigenvector x when multiplied by a matrix A. Ax = λx.

Eigenvector

A non-zero vector x that, when multiplied by a matrix A, results in a vector parallel to the original vector x. Ax = λx.

Determinant of a Matrix

A value calculated from a square matrix that provides information about the transformation the matrix represents. Determinants are often used to solve systems of equations.

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Inverse of a Matrix

A matrix A⁻¹ that, when multiplied by A, equals the identity matrix (I). It essentially reverses the transformation.

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Adjugate Matrix

A matrix used in the formula for finding the inverse of a matrix, calculated using cofactors and the determinant.

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Cofactor

A signed minor determinant used to compute elements in the adjugate of a matrix and in formulas for the inverse.

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Matrix Transformation

A way to modify vectors by multiplying by a matrix. The result may change the vector's direction or magnitude. Matrices transform vectors.

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Eigenpair

An eigenvalue and its corresponding eigenvector.

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Eigenspace

The set of all eigenvectors corresponding to a particular eigenvalue, plus the zero vector.

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Homogeneous System

A system of linear equations where the constant terms are all zero.

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Non-trivial Solution

A solution to a homogeneous system other than the zero vector.

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Finding Eigenvectors

The process of solving the equation (A - λI)x = 0 for x.

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Identity Matrix

A square matrix with 1s on the main diagonal and 0s elsewhere.

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Singular Matrix

A square matrix that has a determinant of zero. This means it cannot be inverted.

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Determinant of (A - λI)

Determines if eigenvalues exist for a matrix A. The determinant is zero if and only if eigenvalues exist.

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Triangular Matrix

A matrix where all entries either above or below the diagonal are zero.

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Eigenvalues of a Triangular Matrix

The diagonal entries of a triangular matrix are its eigenvalues.

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Augmented Matrix

A matrix representing a system of linear equations. It includes both the coefficient matrix and the constant term.

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Study Notes

Cramer's Rule and Inverse of a Matrix

  • System of Equations: Solve Ax = b, where A is an n x n matrix and x and b are vectors.
  • Cramer's Rule: Calculate each element of x using determinants:
    • xᵢ = det(Aᵢ) / det(A)
    • Aᵢ is the matrix A with the i-th column replaced by b.
  • Example: Given A = [[1, 2], [3, -1]] and b = [[6], [-1]], find x.
    • det(A) = -1 - 6 = -7
    • det(A₁) = (6*-1) - (-1*7) = -6 + 10 = 4
    • det(A₂) = (1*(-1)) - (3*6) = -1 - 18 = -19
    • x₁ = 4 / -7 = -4/7
    • x₂ = -19 / -7 = 19/7

Properties of Determinants

  • Transpose: det(AT) = det(A)
  • Matrix Multiplication: det(AB) = det(A) * det(B)

Formula for the Inverse of a Matrix

  • Adjugate: A-1 = adj(A) / det(A)
  • Cofactors: Cᵢⱼ = (-1)i+j * det(Aᵢⱼ)
    • Aᵢⱼ is the matrix A without the i-th row and j-th column

Eigenvectors and Eigenvalues

  • Definition: An eigenvalue (λ) of a matrix A is a scalar that, when multiplied by a non-zero vector (x), produces a vector that is a scalar multiple of the original vector Ax = λx. The vector x is the eigenvector.

  • Example: If A = [[5, 2], [1, 6]] and x = [[1], [2]], then Ax = [[7], [13]] ; which is a scalar multiple of x (because 7/1 = 13/2)

  • Eigenvector (Example): Let A = [[5, 2], [1, 6]]

    Eigenvector x = [ 1,2] Ax = [7, 13] => eigenvalue = 7 So, Ax=λx

  • Finding Eigenvectors and Eigenvalues: One method to solve for eigenvectors and eigenvalues is to solve the equation: (A-λI)x=0 for x where λ is the eigenvalue, and I is the identify matrix

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