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Q: How do you prove if the determinant of A is not equal to zero then the matrix A is invertible?

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What is "a 3b"? Is it a3b? or a+3b? 3ab? I think "a3b" is the following: A is an invertible matrix as is B, we also have that the matrices AB, A2B, A3B and A4B are all invertible, prove A5B is invertible. The problem is the sum of invertible matrices may not be invertible. Consider using the characteristic poly?

Assume that f:S->T is invertible with inverse g:T->S, then by definition of invertible mappings f*g=i(S) and g*f=i(T), which defines f as the inverse of g. So g is invertible.

The trace of an nxn matrix is usually thought of as the sum of the diagonal entries in the matrix. However, it is also the sum of the eigenvalues. This may help to understand why the proof works. So to answer your question, let's say A and B are matrices and A is similar to B. You want to prove that Trace A=Trace B If A is similar to B, there exists an invertible matrix P such that A=(P^-1 B P) Now we use the fact that Trace (AB)= Trace(BA) for any nxn matrices A and B.This is easy to prove directly from the definition of trace. (ask me if you need to know) So using this we have the following: Trace(A)=Trace(P^-1 B P)=Trace (BPP^-1)=Trace(B) and we are done! Dr. Chuck

If x is a null matrix then Ax = Bx for any matrices A and B including when A not equal to B. So the proposition in the question is false and therefore cannot be proven.

Automated proofs are a complicated subject. If you are not an expert on the subject, all you can hope for is to write a program where you can input a sample matrix (or that randomly generates one), and verifies the proposition for this particular case. If the proposition is confirmed in several cases, this makes the proposition plausible, but is by no means a formal proof.Better try to prove it without writing any program.Note: it is not even true; it is the inverse of the matrix which gives identity when is multiplied with the original matrix.

Let's prove that rho(A)=2-norm(A) for A symmetrical and then prove the relation between 1-norm and 2-norm. Both are easy.

You cannot prove it since it is axiomatic. You can get consistent theories (matrix algebra, for example) where ab is not ba.

Let A be a matrix which is both symmetric and skew symmetric. so AT=A and AT= -A so A =- A that implies 2A =zero matrix that implies A is a zero matrix

You prove that the two sides (not the bases) are equal in length. Or that the base angles are equal measure.

The answer depends on what you mean by equal. Equal in area? Congruent?

Recall that if a matrix is singular, it's determinant is zero. Let our nxn matrix be called A and let k stand for the eigenvalue. To find eigenvalues we solve the equation det(A-kI)=0for k, where I is the nxn identity matrix. (<==) Assume that k=0 is an eigenvalue. Notice that if we plug zero into this equation for k, we just get det(A)=0. This means the matrix is singluar. (==>) Assume that det(A)=0. Then as stated above we need to find solutions of the equation det(A-kI)=0. Notice that k=0 is a solution since det(A-(0)I) = det(A) which we already know is zero. Thus zero is an eigenvalue.

You cannot prove that because it's false

prove any two adjacent triangles as congruent

It isn't equal, and any proof that they are equal is flawed.

Base angles are equal

a parallelogram with opposite equal acute angles, opposite equal obtuse angles, and four equal sides.

Corresponding angles must be equal; in this case, that would be angle f. To prove that the two triangles are equal, you would have to prove that at least another pair of angles is also equal, for example, angle b equal to angle d. Or prove some other facts, like the ratio between certain corresponding sides.

It is not true so you cannot prove it. You can concoct a "proof" that might look OK but it will be flawed.

to prove cash you look at the amount of money you have and accounting books. if the value is equal then you have proved cash

to prove cash you look at the amount of money you have and accounting books. if the value is equal then you have proved cash

No, but there is a way to prove that zero equals one.

It is a fact - by definition. You cannot prove it. You can verify it by comparing the two masses.

use you brain

It doesn't...I thought that was clear enough...

Say that the angles all equal 90 degrees and that all sides are equal in length.