| № |
Condition |
free/or 0.5$ |
| m96333 | Show by a column (row) expansion that if
Is upper (lower) triangular, then det(A) = (11(22 . . . (nn. |
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| m96342 | Show that (1 - 4x)-1/2 generates the sequence (2nn), n ∈ N. |
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| m96343 | Show that {1, cosx, cos(2x), cos(3x),...} is an orthogonal set in C[0, π] with respect to the inner product (f, g) = ∫n0 f(x)g(x)dx. |
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| m96348 | Show that 2a - 0 in Z10 holds in Z10 if and only if a = 0 or a = 5. |
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| m96353 | Show that
(a) AB + AC = A(B + C); and
(b) BA + CA = (B + C)A.
[In general, if A is an m × n matrix and B, C are n × p matrices, then AB + AC = A(B + C). For n × p matrices B, C and a p × q matrix A, it follows that BA + CA = (B + C)A. These two results are called the Distributive Laws for Matrix Multiplication over Matrix Addition.]
Let |
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| m96356 | Show that (a) fails if normal is replaced by hermitian. |
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| m96359 | Show that a set S = {v1, v2, ( ( ( ( vn} of vectors in Rn (Rn) spans Rn (Rn) if and only if the rank of the matrix whose jth column (jth row) is vj is n. |
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| m96360 | Show that a square matrix which has a row or a column consisting entirely of zeros must be singular. |
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| m96375 | Show that each of the following arguments is invalid by providing a counterexample - that is, an assignment of truth values for the given primitive statements p, q, r, and 5 such that all premises are true (have the truth value 1) while the conclusion is false (has the truth value 0).
(a) [(p ∧ ¬q) ∧ [p → (q → r)]] → ¬r
(b) [[(p ∧ q) → r] ∧ (¬q ∨ r)] → p
(c)
(d) |
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| m96376 | Show that each of the following transition matrices reaches a state of equilibrium?
(a)
(b)
(c)
(d) |
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| m96377 | Show that every vector in R3 of the form
For t any real number, is in the null space of the matrix |
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| m96382 | Show that for all integers n, r ≥ 0, if w + 1 > r, then |
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| m96385 | Show that for primitive statements p, q,
p ∨ q ⇔ [(p ∧ ¬ q)] ∨ (¬p ∧ q)] ⇔ ¬ (p ↔ q). |
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| m96389 | Show that if A and B are orthogonal matrices, then AB is an orthogonal matrix? |
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| m96390 | Show that if A and B are similar matrices, then Ak and Bk are similar for any positive integer k. |
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| m96391 | Show that if A and B are similar, then AT and BT are similar. |
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| m96392 | Show that if A and B are square matrices, then det
= (det A)(det B). |
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| m96393 | Show that if A , B, and C are square matrices, then det
= (det A) (det B). |
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| m96394 | Show that if A is a matrix such that in each row and in each column one and only one element is not equal to 0, then det(A) ( 0. |
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| m96395 | Show that if A is a nonsingular matrix, then adj A is nonsingular and |
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