Principles and Short Q's

Question 1

Describe the shape of the equation x1 + x2 + x3 = 0.

Answer 1

A plane through (0,0,0)

Question 2

T or F. If the constants of an SLE are all zero than the SLE is consistent?

Answer 2

True

Question 3

T or F. If 3 lines in the xy-plane form a triangle, then the SLE corresponding to the equations of these lines has 3 solutions, one for each vertex of the triangle.

Answer 3

False

Question 4

Is A^TA symmetric?

Answer 4

Yes

Question 5

T or F. If A is a square matrix and A^2 has an all zero column then A has an all zero column.

Answer 5

False

Question 6

T or F. If A is invertible than A^T is invertible and has inverse (A^-1)^T.

Answer 6

True

Question 7

Formula for number of expressions for determinant of n x n matrix with cofactor expansions.

Answer 7

n x 2 expressions.

Question 8

T or F. Let A and B be a pair of n x n invertible matrices. Then |A+B| = |A| + |B|.

Answer 8

False.

Question 9

T or F. Let A be a square matrix. Then there exists a lower triangular matrix L and an upper triangular matrix U such that A = LU.

Answer 9

False

Question 10

T or F. A matrix is invertible if and only if it is the product of elementary matrices.

Answer 10

True

Question 11

Let A be a square matrix and suppose there exists a non-zero vector x such that Ax = 0. Then A is not invertible

Answer 11

True

Question 12

T or F. Every matrix can be reduced by ERO’s to a unique RREF matrix.

Answer 12

True

Question 13

What are the five properties of matrix arithmetic (with examples)?

Answer 13

1. Matrix addition is commutative: A + B = B + A
2. Matrix addition is associative: (A + B) + C = A + (B + C)
3. Matrix multiplication is not commutative: AB != BA
4. Matrix multiplication is associative: (AB)C = A(BC)
5. Distributive Laws for matrix multiplication over matrix addition. (A + B)C = AC + BC A(B+C) = AB + AC

Question 14

What are the four transpose principles?

Answer 14

A = A^T if A is symmetric
(A^T)^T = A
(AB)^T = B^TA^T
AA^T and A^TA are square, often have different sizes and are symmetric.

Question 15

What are the three square matrix principles if two matrices A and B are square matrices?

Answer 15

A + B
AB
BA are all square matrices

Question 16

If AB = I and BA = I. What is B?

Answer 16

B is A^-1

Question 17

What is the inverse of the identity?

Answer 17

The identity.

Question 18

If A is invertible what does AB equal for any matrix B != 0?

Answer 18

AB != 0
Similarly, BA != 0

Question 19

What does singular, non-singular mean?

Answer 19

If A is not invertible; A is singular
If A is invertible; A is non-singular

Question 20

T or F. If x i is a leading variable of a homogenous SLE then x i can be expressed solely in terms of the free variables x j for which j > i.

Answer 20

True.

Question 21

What is the formula for L if U is found in an LU decomposition of the matrix A.

Answer 21

EkEk-1..E1A = U
A = E1^-1…Ek-1^-1 Ek^-1 U

Question 22

Express A^-1 as a product of k elementary matrices.

Answer 22

EkEk-1….E1

Question 23

If A is invertible what can be said about its inverse.

Answer 23

It is unique

Question 24

Formula for determinant of A = (a b)
(c d)

Answer 24

det A = ad - bc

Question 25

Formula for inverse.

Answer 25

A^-1 = 1 / det(A) * adj (A)

Question 26

Formula for adj (A) for 2 x 2 matrix A = (a b) (c d)

Answer 26

= (d -b) (- c a)

Question 27

Let A,B be square and invertible matrices. What can be said about their product.

Answer 27

If AB is invertible, than (AB)^-1 = B^-1A^-1

Question 28

How many ways are there to calculate the determinant of an n x n matrix?

Answer 28

2 x n

Question 29

Formula for calculating det(A) for matrices greater than n = 2.

Answer 29

A * adj (A) = adj(A) * A = det(A) * I

Question 30

If matrix B is achieved from a single row switch of matrix A, what can we say about det(B)?

Answer 30

det(B) = -(det(A))

Question 31

If matrix B is achieved from adding a scalar multiple of one row to another, what can we say about det(B)?

Answer 31

det(B) = det(A)

Question 32

If matrix B is achieved from multiplying a row of A by a scalar constant, what can we say about det(B)?

Answer 32

det(B) = scalar constant * det(A)

Question 33

What is det(AB)?

Answer 33

det(A)det(B)

Question 34

What is det(kA) for n x n matrix A?

Answer 34

k ^ n * (det (A))

Question 35

What four conditions mean det(A) = 0 for any matrix A?

Answer 35

1. If A has an all zero row or column 2. If any two rows or columns of A are equal 3. If any row is a scalar multiple of another row or if any column is a scalar multiple of another column. 4. If the sum of all of the rows except one equals the remaining row or if the sum of all of the columns except one equals the remaining column.

Question 36

If A is invertible what can be assumed about det(A)? Formula for det(A^-1) in relation to det(A). Formula for det(A ^ T) in relation to det(A).

Answer 36

det(A) != 0 det(A^-1) = 1 / det(A) det(A^T) = det(A)

Question 37

Values of det(E) for different elementary matrices E.

Answer 37

row-switch: -1 multiply by scalar: scalar add scalar multiple of another row: 1

Question 38

What can be said about all elementary matrices and their inverse?

Answer 38

They are all invertible and their inverses are elementary matrices.

Question 39

If LU is a decomposition of A, what is the relationship between the determinants of A, L and U?

Answer 39

det(A) = det(L) * det(U)

Question 40

Define row equivalent matrices.

Answer 40

Row-equivalent matrices A and B have some invertible square matrix X such that B = XA or if one can be obtained from the other by ERO's

Question 41

What are the six equivalent statements about an n x n matrix A? i.e. what six statements are all true when one of them is true.

Answer 41

1. A is invertible. 2. For every vector of size n, the equation Ax = v has a unique solution. 3. For every vector of size n, the equation Ax = v has a solution. 4. The equation Ax = 0 has only the trivial solution 0. 5. There are no free variables in the RREF form of A. 6. The RREF of A is I, the identity matrix.

Question 42

What does it mean to say a matrix A is homogenous?

Answer 42

Ax = b where b is the zero factor.

Question 43

What is the trivial solution?

Answer 43

Ax = b where x = 0.

Question 44

What is the nullspace of A?

Answer 44

The set of all solutions of the homogenous system Ax = 0 is called the nullspace of A

Question 45

Of matrix A and B have distinct RREF matrices, what can be said about their solution sets.

Answer 45

They have distinct solution sets.

Question 46

For matrix A and matrix B what can be said about their solution sets if they are row equivalent. What can be said about their RREF forms and rank?

Answer 46

They have identical solution sets. RREF's are identical Ranks are the same

Question 47

Define Cramer's Rule.

Answer 47

Let A be an invertible n x n matrix. Let b be a vector of length n. The solution to Ax = b is unique and is given by xi = |Ai|/|A|, where Ai is obtained from A by replacing the ith column of A with the column vector b.