# Proving B^2=B & Finding det(B) | Linear Algebra Exercises

• hola
In summary, B is a n*n matrix that either has det(B) =1 or is singular. If Transpose(B) = B^-1, det(B) is det(B^-1).
hola
B is a n*n matrix

1. Let B^2 =B. Prove that either det(B) =1 or B is singular.
2. If Transpose(B) = B^-1 , what is det(B)?

Sorry I am asking, but I can't figure them out! I'd really like to improve my linear algebra skills.
Thanks!

The first is really easy.

HINT:What is $\mbox{det} \ A\cdot B$ equal to...?(A,B matrices n*n).Then take A=B and recover the result u were supposed to prove.

For the second HINT:multiply to the left by B and use the "det" property u used for 1.

Daniel.

I don't understand, more depth please?

Alright.Take two n*n matrices A and B.Each of them has a determinant.Question:what is the product of their determinants...?

$$\mbox{det} \ A\cdot B =...?$$

Daniel.

det(a*b) = det(a)det(b)

So for #1, I'm still stuck.
What do I do?

the point is that every matrix satisfies a polynomial, and that polynomial tells you rthe eigenvaleus, which tell you whetehr it is singualr or not.

now a polynomial satisfied by a matrix such that A^2 = A would be?

hola said:
det(a*b) = det(a)det(b)

So for #1, I'm still stuck.
What do I do?

Perfect,then u must agree that

$$\mbox{det} \ B^{2}=\left(\mbox{det} \ B\right)^{2}$$ (1)

And now apply "det" on the equation

$$B^{2}=B$$(2)

and use (1) to get a quadratic algebraic eq. in $\mbox{det} \ B$.

Daniel.

dextercioby, I didn't quite get your each of your steps in the last post.

Could you help me in writing the full proof for 1 and 2?
Aaargh! I feel so frustrated. I should have take regular linear algebra instead of the honors one. I suck at proofs.

I'm really sorry, but I need to solve these problems, but can't get them. Thanks!

## 1. What is the definition of "Proving B^2=B"?

Proving B^2=B means showing that a matrix B multiplied by itself (B^2) is equal to the original matrix B. In other words, proving that B is idempotent.

## 2. How do you prove B^2=B?

To prove B^2=B, you can use the properties of matrix multiplication. First, multiply B^2 and then compare it to the original matrix B. If they are equal, then B^2=B is proven. Another way to prove this is by using the fact that for an idempotent matrix, B^2 is equal to its eigenvalues multiplied by the identity matrix.

## 3. What is the purpose of finding det(B)?

The determinant (det) of a matrix B is a scalar value that represents the scaling factor of the matrix. It is useful in solving systems of linear equations, finding the inverse of a matrix, and determining whether a matrix is invertible or singular.

## 4. How do you find det(B)?

The determinant of a matrix B can be found by using the cofactor expansion along any row or column. Another method is by using the Laplace expansion, which involves breaking down the matrix into smaller matrices and calculating the determinants of those matrices.

## 5. Can B^2=B and det(B) be used interchangeably to prove that a matrix is idempotent?

No, B^2=B and det(B) cannot be used interchangeably to prove that a matrix is idempotent. While both properties are related to idempotent matrices, they are different concepts. B^2=B shows that the matrix is equal to its own square, while det(B) only gives information about the scaling factor of the matrix.

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