# Solving Matrix Equations: Inverse of nxn (n=2)

• DiamondV
In summary, when dealing with matrix multiplication, it is important to remember that it is not commutative but it is associative. This means that when solving equations involving matrices, you must multiply both sides of the equation to the same side in order to maintain the proper order of the matrices. In the given example, B^-1AB = C, in order to get rid of the B^-1 on the left, you must multiply both sides by B on the left. Then, to get rid of the B on the right, you must multiply both sides by B^-1 on the right. This results in the equation A = BCB^-1.
DiamondV

## Homework Equations

Inverse of an (nxn) (n=2 only) square matrix:

## The Attempt at a Solution

The answer provided in the solutions does the exact same thing except, where my ?? are. It does A = BCB^-1. Where as I do A = CBB^-1. When I was doing this question I was wondering the same thing. I know matrix multiplication isn't associative( AB isn't equal to BA), so how do I know which way to form the equations? I mean how am I meant to know whether its meant to be A=CBB^-1 or BCB^-1?[/B]

DiamondV said:

## Homework Equations

Inverse of an (nxn) (n=2 only) square matrix:

## The Attempt at a Solution

The answer provided in the solutions does the exact same thing except, where my ?? are. It does A = BCB^-1. Where as I do A = CBB^-1. When I was doing this question I was wondering the same thing. I know matrix multiplication isn't associative( AB isn't equal to BA), so how do I know which way to form the equations? I mean how am I meant to know whether its meant to be A=CBB^-1 or BCB^-1?[/B]
Matrix multiplication is associative. You probably meant to write that matrix multiplication isn't commutative.

Since matrix multiplication isn't commutative, when you multiply one side of an equation to the left, you must multiply the other side of the equation to the left as well. Same for multiplying to the right.
So if ##X=Y##, then ##AX=AY## and ##XA=YA##, but not necessarily ##XA=AY##.
(Here ##X,\ Y, \ A## are square matrices of the same dimension.)

Now in your case: apply this rule to ##B^{-1}AB=C## to get ##A=BCB^{-1}##.

Samy_A said:
Matrix multiplication is associative. You probably meant to write that matrix multiplication isn't commutative.

Since matrix multiplication isn't commutative, when you multiply one side of an equation to the left, you must multiply the other side of the equation to the left as well. Same for multiplying to the right.
So if ##X=Y##, then ##AX=AY## and ##XA=YA##, but not necessarily ##XA=AY##.
(Here ##X,\ Y, \ A## are square matrices of the same dimension.)

Now in your case: apply this rule to ##B^{-1}AB=C## to get ##A=BCB^{-1}##.
Ah. so essentially whatever order they are in the at the left is the same order they get applied to on the right?

DiamondV said:
Ah. so essentially whatever order they are in the at the left is the same order they get applied to on the right?

No: they do not have the same order on the two sides.Look again, carefully.

Ray Vickson said:
No: they do not have the same order on the two sides.Look again, carefully.

Not understanding it completely. So say if I multiply the left side by some variable X and I put it to the left of whatever is already there, I have to do the same to the right? like say A=C, would be XA = XC?

DiamondV said:
Not understanding it completely. So say if I multiply the left side by some variable X and I put it to the left of whatever is already there, I have to do the same to the right? like say A=C, would be XA = XC?
Yes, this is correct. If A=C, then XA=XC.

Samy_A said:
Yes, this is correct. If A=C, then XA=XC.

Ah. Thanks a lot!

You are given that $B^{-1}AB= C$. Knowing that matrix multiplication is not commutative, get rid of the "$B^{-1}$" on the left by multiplying, by B, on both sides on the left: $B(B^{1}AB)= BC$. Because matrix multiplication is "associative" that is the same as $(BB^{-1})AB= AB= BC$. And to get rid of the "B" or the right, multiply on both sides by $B^{-1}$ on the right to get $(AB)B^{-1}=A(BB^{-1})= A= BCB^{-1}$. It's just a matter of keeping track of which side you are on!

## 1. What is a matrix equation?

A matrix equation is an equation that involves matrices, which are rectangular arrays of numbers or variables. It is written in the form of Ax = b, where A is a matrix, x is a column vector, and b is a constant vector.

## 2. What is an inverse matrix?

An inverse matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. It is denoted as A^-1, and is used to solve matrix equations by multiplying both sides of the equation by the inverse matrix.

## 3. How do you find the inverse of a 2x2 matrix?

To find the inverse of a 2x2 matrix, first calculate the determinant of the matrix (ad - bc). Then, switch the positions of the elements on the main diagonal (a and d) and change the signs of the elements on the off-diagonal (b and c). Finally, divide each element by the determinant to get the inverse matrix.

## 4. Can a 2x2 matrix have more than one inverse?

No, a 2x2 matrix can only have one inverse. If the determinant is 0, the matrix does not have an inverse. If the determinant is not 0, the inverse is unique and can be calculated using the method described in the previous answer.

## 5. How is the inverse of a 2x2 matrix used to solve equations?

The inverse of a 2x2 matrix is used to solve equations by multiplying both sides of the equation by the inverse matrix. This removes the matrix from the left side of the equation, leaving x on its own. The resulting equation is then solved for x using basic algebra.

• Precalculus Mathematics Homework Help
Replies
25
Views
1K
• Precalculus Mathematics Homework Help
Replies
18
Views
2K
• Precalculus Mathematics Homework Help
Replies
7
Views
2K
• Precalculus Mathematics Homework Help
Replies
69
Views
4K
• Precalculus Mathematics Homework Help
Replies
9
Views
2K
• Precalculus Mathematics Homework Help
Replies
32
Views
1K
• Precalculus Mathematics Homework Help
Replies
4
Views
2K
• Precalculus Mathematics Homework Help
Replies
4
Views
1K
• Precalculus Mathematics Homework Help
Replies
5
Views
1K
• MATLAB, Maple, Mathematica, LaTeX
Replies
2
Views
1K