Questions and Answers of Applied Linear Algebra

Graph the following planes and determine whether they have a common intersection:x + y + z = 1, x + y = 1, x + z = 1.
Which is faster: Back Substitution or multiplying a matrix by a vector? How much faster?
Find a formula for the transposed product (ABC)T in terms of AT ,BT and CT.
True or false: A singular matrix cannot be regular.
True or false: Every square matrix A commutes with its transpose AT.
(a) Write down the 5 × 5 identity and zero matrices. (b) Write down their sum and their product. Does the order of multiplication matter?
(a) Let U be a m × n matrix and V an n × m matrix, such that the m × m matrix Im + U V is invertible. Prove that In +V U is also invertible, and is given by(b) The Sherman–Morrison–Woodbury
True or false: A square matrix that has a column with all 0 entries is singular. What can you say about a linear system that has such a coefficient matrix?
An alternative solution strategy, also called Gauss–Jordan in some texts, is, once a pivot is in position, to use elementary row operations of type #1 to eliminate all entries both above and below
(a) Show that if A has size n × n, then det(−A) = (−1)n detA. (b) Prove that, for n odd, any n × n skew-symmetric matrix A = −AT is singular. (c) Find a nonsingular skew-symmetric
Prove that if A is an invertible matrix, then AAT and ATA are also invertible.
Write out the following diagonal matrices: (a) Diag(1, 0,−1), (b) Diag(2,−2, 3,−3).
Find the rank of the n × n matrix 1 n+1 2n +1 2 n+2 2n+2 n²n+1 n²n+2 3 n +3 2n + 3 n 2n 3n n 2
True or false: (a) The product of two tridiagonal matrices is tridiagonal.(b) The inverse of a tridiagonal matrix is tridiagonal.
What does it mean if a linear system has a coefficient matrix with a column of all 0’s?
Show that if a square matrix A satisfies A2 − 3A + I = O, then A − 1 = 3I − A.
True or false: One can find an m × n matrix of rank r for every 0 ≤ r ≤ min {m, n}.
True or false: Every m × n matrix has (a) Exactly m pivots; (b) At least one pivot.
Given the factorization explain, without computing, which elementary row operations are used to reduce A to upper triangular form. Be careful to state the order in which they should be applied.
A square matrix P is called idempotent if P2 = P. (a) Find all 2 × 2 idempotent upper triangular matrices. (b) Find all 2 × 2 idempotent matrices.
(a) Let P and Q be n × n permutation matrices and v ∈ Rn a vector. Under what conditions does the equation P v = Qv imply that P = Q? (b) Answer the same question when PA = QA, where A is an
Show that if A is a nonsingular matrix, so is everyy power An.
(a) Let A be an m × n matrix and let M = A | b be the augmented matrix for the linear system Ax = b. Show that either (i) Rank A = rank M, or (ii) Rank A = rank M − 1.(b) Prove that the
Prove that one cannot produce an elementary row operation of type #2 by a combination of elementary row operations of type #1.
Let A be an m × n matrix. What are the permissible sizes for the zero matrices appearing in the identitie AO = O and OA = O?
True or false: If A,B are square matrices of the same size, thenA2 − B2 = (A + B)(A − B).
What is the effect of permuting the columns of its coefficient matrix on a linear system?
Find a nonzero matrix A ≠ O such that A2 = O.
True or false: The pivots of a nonsingular matrix are uniquely defined.
True or false: If A has a zero entry on its main diagonal, it is not regular.
(a) Suppose A̅ is obtained from A by applying an elementary row operation. Let C = AB, where B is any matrix of the appropriate size. Explain why C̅ = A̅B can be obtained by applying the same
Express as a product of elementary matrices. A Но как
Let A be a matrix and c a scalar. Find all solutions to the matrix equation c A = I.

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