A baker bought some flower. He used 2/5 of the flour to make bread and used the rest to make batches of muffins. If he used 16lb
1 answer:
We know he used 2/5 of the flour for bread, we also know he used 16lbs for bread, that means, if the total amount is "x" lbs, then (2/5)x = 16.
so, it was 40lbs total...now, he used 16lbs for bread, so the rest is for batches of muffins, namely 24lbs.
each batch of muffin uses 2/3 lb, so, how many times 2/3 goes into 24?
so, it was 40lbs total...now, he used 16lbs for bread, so the rest is for batches of muffins, namely 24lbs.
each batch of muffin uses 2/3 lb, so, how many times 2/3 goes into 24?
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Answer:
To Prove: Quadrilateral ABCD is a parallelogram.
Proof: In Δ ABE and ΔCDE
1. AE = EC and BE = ED [ Diagonals bisect each other]
2.∠ AEB = ∠ CED [ vertically opposite angles]
Δ ABE ≅ ΔCDE---------- [SAS]
∠ ACD ≅ ∠CAB [Corresponding angles of congruent triangles are congruent⇒This statement is untrue ∴ these are alternate interior angles not corresponding angles.]
6. The converse of alternate interior interior angle theorem states that if two parallel lines are cut by a transversal then alternate interior angles are equal.
7. In ΔBEC and ΔAED
∠BEC = ∠AED [ Vertical Angles Theorem ]
AE = EC and BE = ED [ Diagonals bisect each other]
⇒ ΔBEC≅ ΔDEA [ SAS criterion for congruence]
9. DBC ≅ BDA [ Corresponding angles of congruent triangles are congruent⇒This statement is untrue ∴ these are alternate interior angles not corresponding angles.]
As pair of triangles are congruent ∵ quadrilateral ABCD is a parallelogram.
Step 3 is m∠AEB = m∠CED
These pair of angles are vertically opposite angles of ΔAEB and ΔCED.
Option [D. Vertical Angles Theorem] is correct.
Answer:
the P matrix you are looking for is P=(1/) · [[1 1 0 0],[1 -1 0 0],[0 0 1 1],[0 0 1 -1]]
Step-by-step explanation:
Answer:
For an orthogonal diagonalization of any matrix you have to:
1º) Find the matrix eigenvalues in a set order.
2º) Find the eigenvectors of each respective eigenvalues.
Tip: You can write the matrix A like A = D P
3º) D is the diagonal matrix with each eigenvalue (in order) in the diagonal.
4º) Write P as the normalized eigenvectors in order (in columns).
Tip 2: Remember, ·P = I, so if A =
D P, then:
P A = P
D P
= I D I = D
So the P we are looking for is the of the diagonalization.
Tip 3: In this case, A is a block matrix with null nondiagonal submatrixes, therefore its eigenvalues can be calculated by using the diagonal submatrixes. The problem is reduced to calculate the eigenvalues of A₁₁ = A ₂₂ = [[5 3],[3 5]]
Solving:
1º)the eigenvalues of A₁₁ are {8,2}, therefore the D matrix is
2º) the eigenvectors of A₁₁ are P₈= {[1 1]T} P₂= {[1 -1]T}, therefore normalizing the eigenvectors you obtain P = (1/) · [[1 1 0 0],[1 -1 0 0],[0 0 1 1],[0 0 1 -1]] (you can see that P =
in this case).
As said in "tip 2": P A = P
D P
= I D I = D
So the P obtained is the one you are looking for.