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 (correct)

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

4 (correct)

-4

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?

The system must be homogeneous. Signup and view all the answers

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

(A - λ)x = 0 Signup and view all the answers

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

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

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

Az does not equal a scaled version of z. Signup and view all the answers

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

Replace column # i of A with b Signup and view all the answers

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

x2 = det(A2)/det(A) Signup and view all the answers

What property of determinants is established by Theorem 5?

Determinants of a matrix and its transpose are equal Signup and view all the answers

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

det(AB) = det(A) * det(B) Signup and view all the answers

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

The determinant of the matrix A without row i and column j Signup and view all the answers

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

det(A) must be non-zero Signup and view all the answers

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

A scalar such that Ax = λx for some non-zero vector x Signup and view all the answers

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

Eigenvectors remain unchanged under the transformation by A Signup and view all the answers

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

det(A - λI) = 0 Signup and view all the answers

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

They are eigenvectors corresponding to λ = 2. Signup and view all the answers

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

2x1 - x2 + 6x3 = 0 Signup and view all the answers

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

The eigenvalues are the diagonal entries themselves. 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)?

(a11 - λ)(a22 - λ)(a33 - λ) Signup and view all the answers

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

It is the span of all possible eigenvectors for that eigenvalue. 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?

Variables that can take any value without restriction. Signup and view all the answers

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

A - λI is singular if and only if λ is an eigenvalue of A. Signup and view all the answers

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(A^T) = 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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Description

Explore key concepts in linear algebra, including Cramer's Rule for solving systems of equations and the calculation of matrix inverses. This quiz covers determinants, properties of determinants, and the basics of eigenvectors and eigenvalues. Test your understanding of these fundamental topics in matrix theory.