Final Exam (need to memorize)

- Properties of determinants - Formulas - Conditions for methods - things to remember

What is another way to express the following determinants?
det(u, v+w)
det (u+v, w)

det(u, v) + det(u, w)
det(u, w) + det(v, w)

What is another way to express the following determinant?
det(u, kv) where k is a scalar multiple.

k det(u, v) = det(ku, v)

How does swapping columns/rows affect the determinant?

det(u, v)

The determinant of the permutation (det(P)) is…

• positive if even row exchanges

• negative if odd row exchanges

- det(v, u)

How can the following matrix be rewritten?

| ta tb |
| c d |

t * | a b | = t * det(matrix)
| c d |

There is linearity in the rows

What is the determinant of a matrix that has 2 equal rows or columns?

zero

What happens to the determinant of a matrix when…

• subtract l * row i from row k

The determinant of the matrix doesn’t change, since the cofactor of the mutation evaluates to zero

What is the determinant of a matrix with a row of zeros?

zero

What is the determinant of an upper/lower triangular or diagonal matrix?

The determinant is the product of the diagonal entries

• product of pivots assuming no row exchanges

What is the determinant of a matrix A that is not invertible (singular)?

zero

• A matrix is invertible if it’s determinant does NOT equal zero

What is the det(AB)

• A and B are two different matrices

det(A) * det(B)

What is the det(A^-1)

• A inverse

A(inverse) * A = Identity… therefore

• det(A(inv)) * det(A) = 1 (det of identity)

• det(A(inv)) = 1 / det(A)

What is the following determinant?

det(A²)

det(A²) = det(AA) = det(A) * det(A)

What is the following determinant?

det(2A)

det(2A) = 2^{n _*} det(A)

What is the following determinant?

det(A^T)

det(A)

What is Cramers rule?

• Conditions

• Formula

Given a system Ax = b, Cramer’s rule gives a formula to find each component of x, such that

Cramer’s rule is only applicable IF A is an invertible matrix (det does not equal 0)

Matrix B_i is a matrix in which the ith column of A is replaced with the vector b and every other column is maintained

• x_i = det(B_i) / det(A)

What is the formula for the inverse of matrix A?

Each entry of the C (cofactor) matrix is given by…

• c_ij = (-1)^{i + j} det(M_ij)

• M is the cofactor matrix

Let A, B, and C be three points in R²

What is the area of a parallelogram determined by vectors AB. and AC?

What is the area of the triangle formed by ABC?

What is the volume of the parallelepiped?

• Let A, B, C, and D be three points in R³

What is the volume of the pyramid or tetrahedron?

• Let A, B, C, and D be three points in R³

How do you find an eigenvalue(s)?

• solve the characteristic equation det(A - λI) = 0

• Each root is an eigenvalue of the matrix A

How do you find eigenvectors?

• find linearly independent vectors (a basis) of λ-eigenspace N(A - λI) for each eigenvalue λ

• The geometric multiplicity of an eigenvalue λ is dim N(A - λI), which is the number of linearly independent eigenvector associated with λ

How can the determinate of a matrix be expressed with eigenvalues?

det (A) = product of all eigenvalues

What can be said about the eigenvalues of a matrix that is NOT invertible?

zero is an eigenvalue of the matrix

the number of non-zero eigenvalues is the same as the rank of the matrix A

What are the eigenvalues of A^k ?

(eigenvalues of A)^k

What are the eigenvalues of cA + dI ?

c(eigenvalue of A) + d for each eigenvalue of A

When is the matrix A (n x n) diagonalizable?

If and only if

• A has n linearly independent eigenvectors (x1, x2, . . . , xn)

When are two matrices A and B similar?

• In the context of diagonalization

If there exists and invertible matrix C such that…

• A = CBC^-1

These two matrices have the same eigenvalues (hence the same trace and determinants)

The Spectral Theorem

Every real symmetric matrix S can be factors into the form

• QΛQ^T, where Q is a real orthogonal matrix and Λ is a real diagonal matrix Λ

• Find the eigenvalues of the matrix S

• Find the eigenvectors to each eigenvalue value

  • If any two eigenvectors stem from the same eigenvalue… use gram Schmidt to find orthogonal vector of the two

• Normalize the eigenvectors (divide eigenvectors by magnitude of eigenvector)

• Each normalized eigenvector is in matrix Q and the diagonal matrix is made up of eigenvalues of S

What are the 5 tests for positive definiteness of a (symmetric) n x n matrix S?

1. All eigenvalues of S are positive

2. x^TSx > 0 for any non-zero vector x in Rn

3. S = A^TA for a full column rank matrix A (Cholesky Factorization)

4. All D_k > 0, where D_k is the determinant of the left-upper k by k submatrix of A

5. S has n pivots and all pivots are positive

What are the five tests for a positive semi definite (symmetric) n x n matrix S?

1. All eigenvalues of S are non-negative

2. x^TSx >= 0 for any vector x in Rn

3. S = A^TA for a matrix A (Cholesky Factorization)

4. All D_k >= 0, where D_k is the determinant of the left-upper k by k submatrix of A

5. All pivots of S are positive