Algebraic manipulation of identity matrix

In summary, the conversation discusses confusion regarding the manipulation of the identity matrix in a practice question. The solution manual showed two different ways of solving for A in the equation I - 2A = B, leading to the question of whether the two statements are equivalent. It is clarified that both statements are equivalent due to the properties of matrix addition and scalar multiplication. No special manipulations are needed to bring the identity matrix to the other side.
  • #1
KataKoniK
1,347
0
I was doing a few practice questions and was a bit confused about how the solution manual manipulated the I identity matrix. For example,

Just say you had the following

I - 2A = B

where A and B are 2x2 matrices. B is given, but we need to find A.

Therefore, shouldn't it be

-2A = B - I
A = -(1/2) * (B - I)?

Because the book did it like this

2A = I - B
A = (1/2) * (1 - B)

Can anyone tell me why? Because both statements do not seem to be equivalent. (unless I'm missing something)

Thanks in advance.
 
Physics news on Phys.org
  • #2
I think you may have had one of those brain lapses that I hate so much. The statements are equivalent, as the negative sign will distribute.

-(1/2)*(B-I) = (1/2)*-(B-I) = (1/2)*(I-B)
 
  • #3
Argh, thanks a bunch. So it's just basic algebraic manipulations correct? No special things I have to do to bring I to the other side?
 
  • #4
Nope, nothing special. Matrix addition and scalar multiplication satisfy all of the properties of regular addition and multiplication (commutativity, associativity, distributivity, etc.), so basic manipulations will suffice.
 
  • #5
Thanks a lot!
 

What is the identity matrix?

The identity matrix, also known as the unit matrix, is a square matrix with 1s on the main diagonal and 0s everywhere else. It is denoted by the symbol I or In, where n represents the size of the matrix.

Why is the identity matrix important in algebraic manipulation?

The identity matrix serves as the multiplicative identity for matrices. This means that when the identity matrix is multiplied by any other matrix, the result is the original matrix. This property is crucial in algebraic manipulation as it allows us to simplify equations and solve for unknown variables.

How is the identity matrix used in matrix operations?

The identity matrix is used in matrix operations as a way to preserve the properties of the original matrix. For example, when multiplying a matrix by the identity matrix, the original matrix remains unchanged. Similarly, when performing matrix addition or subtraction, the identity matrix can be added or subtracted without changing the result.

Can the identity matrix be of any size?

Yes, the identity matrix can be of any square size. For example, a 3x3 identity matrix would have 1s on the main diagonal and 0s everywhere else, while a 4x4 identity matrix would have 1s on the main diagonal and 0s everywhere else. The size of the identity matrix depends on the size of the matrix it is being multiplied by.

How does the identity matrix relate to the inverse of a matrix?

The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. This means that the identity matrix is the inverse of itself. Inverse matrices are useful in solving equations involving matrices, as multiplying by the inverse results in the original matrix being cancelled out.

Similar threads

  • Linear and Abstract Algebra
Replies
2
Views
1K
Replies
7
Views
782
  • Linear and Abstract Algebra
Replies
8
Views
993
Replies
34
Views
2K
  • Linear and Abstract Algebra
Replies
9
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
878
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
957
  • Precalculus Mathematics Homework Help
Replies
25
Views
829
  • Precalculus Mathematics Homework Help
2
Replies
57
Views
3K
Back
Top