Principles and Short Q's Flashcards
Describe the shape of the equation x1 + x2 + x3 = 0.
A plane through (0,0,0)
T or F. If the constants of an SLE are all zero than the SLE is consistent?
True
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.
False
Is A^TA symmetric?
Yes
T or F. If A is a square matrix and A^2 has an all zero column then A has an all zero column.
False
T or F. If A is invertible than A^T is invertible and has inverse (A^-1)^T.
True
Formula for number of expressions for determinant of n x n matrix with cofactor expansions.
n x 2 expressions.
T or F. Let A and B be a pair of n x n invertible matrices. Then |A+B| = |A| + |B|.
False.
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.
False
T or F. A matrix is invertible if and only if it is the product of elementary matrices.
True
Let A be a square matrix and suppose there exists a non-zero vector x such that Ax = 0. Then A is not invertible
True
T or F. Every matrix can be reduced by ERO’s to a unique RREF matrix.
True
What are the five properties of matrix arithmetic (with examples)?
- Matrix addition is commutative: A + B = B + A
- Matrix addition is associative: (A + B) + C = A + (B + C)
- Matrix multiplication is not commutative: AB != BA
- Matrix multiplication is associative: (AB)C = A(BC)
- Distributive Laws for matrix multiplication over matrix addition. (A + B)C = AC + BC A(B+C) = AB + AC
What are the four transpose principles?
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.
What are the three square matrix principles if two matrices A and B are square matrices?
A + B
AB
BA are all square matrices
If AB = I and BA = I. What is B?
B is A^-1
What is the inverse of the identity?
The identity.
If A is invertible what does AB equal for any matrix B != 0?
AB != 0
Similarly, BA != 0
What does singular, non-singular mean?
If A is not invertible; A is singular
If A is invertible; A is non-singular
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.
True.
What is the formula for L if U is found in an LU decomposition of the matrix A.
EkEk-1..E1A = U
A = E1^-1…Ek-1^-1 Ek^-1 U
Express A^-1 as a product of k elementary matrices.
EkEk-1….E1
If A is invertible what can be said about its inverse.
It is unique
Formula for determinant of A = (a b)
(c d)
det A = ad - bc
Formula for inverse.
A^-1 = 1 / det(A) * adj (A)
Formula for adj (A) for 2 x 2 matrix A = (a b)
(c d)
= (d -b)
(- c a)
Let A,B be square and invertible matrices. What can be said about their product.
If AB is invertible, than (AB)^-1 = B^-1A^-1
How many ways are there to calculate the determinant of an n x n matrix?
2 x n
Formula for calculating det(A) for matrices greater than n = 2.
A * adj (A) = adj(A) * A = det(A) * I
If matrix B is achieved from a single row switch of matrix A, what can we say about det(B)?
det(B) = -(det(A))
If matrix B is achieved from adding a scalar multiple of one row to another, what can we say about det(B)?
det(B) = det(A)
If matrix B is achieved from multiplying a row of A by a scalar constant, what can we say about det(B)?
det(B) = scalar constant * det(A)
What is det(AB)?
det(A)det(B)
What is det(kA) for n x n matrix A?
k ^ n * (det (A))
What four conditions mean det(A) = 0 for any matrix A?
- If A has an all zero row or column
- If any two rows or columns of A are equal
- If any row is a scalar multiple of another row or if any column is a scalar multiple of another column.
- 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.
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).
det(A) != 0
det(A^-1) = 1 / det(A)
det(A^T) = det(A)
Values of det(E) for different elementary matrices E.
row-switch: -1
multiply by scalar: scalar
add scalar multiple of another row: 1
What can be said about all elementary matrices and their inverse?
They are all invertible and their inverses are elementary matrices.
If LU is a decomposition of A, what is the relationship between the determinants of A, L and U?
det(A) = det(L) * det(U)
Define row equivalent matrices.
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
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.
- A is invertible.
- For every vector of size n, the equation Ax = v has a unique solution.
- For every vector of size n, the equation Ax = v has a solution.
- The equation Ax = 0 has only the trivial solution 0.
- There are no free variables in the RREF form of A.
- The RREF of A is I, the identity matrix.
What does it mean to say a matrix A is homogenous?
Ax = b where b is the zero factor.
What is the trivial solution?
Ax = b where x = 0.
What is the nullspace of A?
The set of all solutions of the homogenous system Ax = 0 is called the nullspace of A
Of matrix A and B have distinct RREF matrices, what can be said about their solution sets.
They have distinct solution sets.
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?
They have identical solution sets.
RREF’s are identical
Ranks are the same
Define Cramer’s Rule.
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.
