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sveta [45]
3 years ago
10

Which equation could be used to find the number of minutes, m, in h hours?

Mathematics
1 answer:
Mrrafil [7]3 years ago
7 0
Y=m+h is the equation

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Linear Algebra question! Please help!
kozerog [31]

Answers:

  1. false
  2. false
  3. true
  4. false
  5. True

==================================================

Explanation:

Problem 1

This is false because the A and B should swap places. It should be (AB)^{-1} = B^{-1}A^{-1}.

The short proof is to multiply AB with its inverse (AB)^{-1}  and we get: (AB)*(AB)^{-1} = (AB)*(B^{-1}A^{-1}) = A(B*B^{-1})*A^{-1} = A*A^{-1} = I

The fact we get the identity matrix proves that we have the proper order at this point. The swap happens so that B matches up its corresponding inverse B^{-1} and the two cancel each other out.

Keep in mind matrix multiplication is <u>not</u> commutative. So AB is not the same as BA.

-------------------------

Problem 2

This statement is true if and only if AB = BA

(A+B)^2 = (A+B)(A+B)

(A+B)^2 = A(A+B) + B(A+B)

(A+B)^2 = A^2 + AB + BA + B^2

(A+B)^2 = A^2 + 2AB + B^2 ... only works if AB = BA

However, in most general settings, matrix multiplication is <u>not</u> commutative. The order is important when multiplying most two matrices. Only for special circumstances is when AB = BA going to happen. In general,  AB = BA is false which is why statement two breaks down and is false in general.

-------------------------

Problem 3

This statement is true.

If A and B are invertible, then so is AB.

This is because both A^{-1} and B^{-1} are known to exist (otherwise A and B wouldn't be invertible) and we can use the rule mentioned in problem 1. Make sure to swap the terms of course.

Or you can use a determinant argument to prove the claim

det(A*B) = det(A)*det(B)

Since A and B are invertible, their determinants det(A) and det(B) are nonzero which makes the right hand side nonzero. Therefore det(A*B) is nonzero and AB has an inverse.

So if we have two invertible matrices, then their product is also invertible. This idea can be scaled up to include things like A^4*B^3 being also invertible.

If you wanted, you can carefully go through it like this:

  1. If A and B are invertible, then so is AB
  2. If A and AB are invertible, then so is A*AB = A^2B
  3. If A and A^2B are invertible, then so is A*A^2B = A^3B

and so on until you build up to A^4*B^3. Therefore, we can conclude that A^m*B^n is also invertible. Be careful about the order of multiplying the matrices. Something like A*AB is different from AB*A, the first of which is useful while the second is not.

So this is why statement 3 is true.

-------------------------

Problem 4

This is false. Possibly a quick counter-example is to consider these two matrices

A = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix} \text{ and } B = \begin{bmatrix}-1 & 0\\0 & -1\end{bmatrix}

both of which are invertible since their determinant is nonzero (recall the determinant of a diagonal matrix is simply the product along the diagonal entries). So it's not too hard to show that the determinant of each is 1, and each matrix shown is invertible.

However, adding those two mentioned matrices gets us the 2x2 zero matrix, which is a matrix of nothing but zeros. Clearly the zero matrix has determinant zero and is therefore not invertible.

There are some cases when A+B may be invertible, but it's not true in general.

-------------------------

Problem 5

This is true because each A pairs up with an A^{-1} to cancel out (similar what happened with problem 1). For more info, check out the concept of diagonalization.

5 0
2 years ago
Lauren Rob need to find a decimal equivalent to 39/50 Laura said she could write an equivalent fraction with 100 as the denomina
Lina20 [59]

Answer:

Laura and Rob are correct

Step-by-step explanation:

we have

39/50

1) Rob said he could divide the numerator by the denominator

so

using a calculator

39/50=0.78

Rob is correct

2) Laura said she could write an equivalent fraction with 100 as the denominator to converted into a decimal

so

Multiply by 2/2

(39/50)*(2/2)=78/100=0.78

Laura is correct

therefore

Laura and Rob are correct

5 0
3 years ago
Solving the equation 5x + 4 = 19​
user100 [1]

Step-by-step explanation:

5x = 19 - 4

5x = 15

x = 15/ 5

= 3

hope it helps!

6 0
2 years ago
Read 2 more answers
Pablo graphs a system of equations. One equation is quadratic and the other equation is linear. What is the greatest number of p
Nady [450]
For this case we have the following type of equations:
 Quadratic equation:
 y = ax ^ 2 + bx + c&#10;
 Linear equation:
 y = mx + b&#10;
 We observe that when equating the equations we have:
 ax ^ 2 + bx + c = mx + b&#10;
 Rewriting we have:
 ax ^ 2 + (b-m) x + (c-b) = 0&#10;
 We obtain a polynomial of second degree, therefore, the maximum number of solutions that we can obtain is 2.
 Answer:
 The greatest number of possible solutions to this system is:
 c.2
7 0
2 years ago
Read 2 more answers
Which choices are in the solution set of the inequality below? Check all that
kaheart [24]

Answer: its -0

Step-by-step explanation:

7 0
2 years ago
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