Multiplication/division of matrices and vectors

In summary, it is not possible to isolate A in the equation Ax=b, as it represents N equations with N2 unknowns and A may not be invertible. However, if A is invertible, then the solution is x=A-1×b.
  • #1
Jhenrique
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1) Let A a square matrix, x a colum vector and b another colum vector. So, I want solve for x the following equation: Ax=b
So: x=b÷A = b×A-1 And this is the answer! Or would be this the correct answer x = A-1×b ?

2) Is possible to solve the equation above for A ? How?
 
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  • #2
Jhenrique said:
1) Let A a square matrix, x a colum vector and b another colum vector. So, I want solve for x the following equation: Ax=b
So: x=b÷A = b×A-1 And this is the answer! Or would be this the correct answer x = A-1×b ?

First of all, you shouldn't use the division symbol ÷ for matrices. Why not? Matrices are noncommutative. This is, it can happen that ##AB \neq BA##. The division ÷ is ambiguous in the noncommutative case, because it is unclear whether ##A##÷##B## means ##AB^{-1}## or ##B^{-1}A##. So you should always use the ##B^{-1}## notation.

Anyway, you want to solve ##A\mathbf{x} = \mathbf{b}##. First of all, ##A## might not be an invertible matrix, in which case, you can't always solve this system (and if you can, the solution might not be unique!). If your matrix is invertible, then you can multiply the equation on the left with ##A^{-1}## and you get

[tex]\mathbf{x} = A^{-1}A\mathbf{x} = A^{-1}\mathbf{b}[/tex]

Multiplying the equation on the right doesn't work since you'll get

[tex]A\mathbf{x}A^{-1} = \mathbf{b}A^{-1}[/tex]

We can't simplify the left-hand side due to noncommutativity.

2) Is possible to solve the equation above for A ? How?

If you mean that ##\mathbf{x}## and ##\mathbf{b}## are known, then no. You don't have enough data for a unique solution. That is, there will be many solutions to this problem.
 
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  • #3
Jhenrique said:
2) Is possible to solve the equation above for A ? How?
micromass said:
If you mean that ##\mathbf{x}## and ##\mathbf{y}## are known, then no. You don't have enough data for a unique solution. That is, there will be many solutions to this problem.
The equation that was referred to was Ax = b, so if x and b are known, there is not a unique matrix A.
 
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  • #4
Jhenrique said:
1) Let A a square matrix, x a colum vector and b another colum vector. So, I want solve for x the following equation: Ax=b
So: x=b÷A = b×A-1 And this is the answer!
That is not the answer. The only way what you wrote would make sense is if A is a 1×1 matrix; i.e., a scalar.

Or would be this the correct answer x = A-1×b ?
That's correct -- if A is invertible.
2) Is possible to solve the equation above for A ? How?
Only if A is a 1×1 matrix. For an N×N matrix, where N>1, the answer is no (not uniquely). Ax=b represents N equations, but you have N2 unknowns.
 
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  • #5
Jhenrique said:
How can I reach the same conclusion?
micromass answered this question in post #2.

If Ax = b, and A is invertible, then multiply on the left by A-1.
 
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  • #6
Mark44 said:
The equation that was referred to was Ax = b, so if x and b are known, there is not a unique matrix A.

Aah, thanks you very much! I really need to proof read my posts better. I'll change my post.
 
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  • #7
Mark44 said:
micromass answered this question in post #2.

If Ax = b, and A is invertible, then multiply on the left by A-1.

Yeah! sorry... I not seen, I'm reading now!

Edit: thanks for everyone from topic!

Edit2: So, Independent of the uniqueness, you are saying that is not possible to isolate A in Ax=b ?
 
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What is the difference between multiplication and division of matrices and vectors?

Multiplication of matrices and vectors is a mathematical operation that combines two or more matrices or vectors to form a new matrix or vector. Division of matrices and vectors, on the other hand, is the inverse operation of multiplication, where a matrix or vector is divided by another matrix or vector to obtain the original matrix or vector.

How do you perform multiplication and division of matrices and vectors?

To multiply matrices and vectors, you need to follow certain rules and procedures depending on the type of matrices or vectors involved. Generally, you need to ensure that the number of columns in the first matrix or vector is equal to the number of rows in the second matrix or vector. To divide matrices and vectors, you need to use the inverse of the second matrix or vector and perform the same steps as multiplication.

What are the applications of multiplication and division of matrices and vectors?

Multiplication and division of matrices and vectors are widely used in various fields such as physics, engineering, computer science, and economics. They are used to solve systems of linear equations, perform transformations in 3D graphics, analyze data in statistics, and much more.

What are the properties of multiplication and division of matrices and vectors?

The properties of multiplication and division of matrices and vectors include commutative, associative, and distributive properties. These properties allow for the rearrangement and simplification of expressions involving matrices and vectors, making it easier to solve complex problems.

What are some common mistakes made when performing multiplication and division of matrices and vectors?

Some common mistakes made when performing multiplication and division of matrices and vectors include forgetting to check for the compatibility of matrices and vectors, confusing the order of multiplication or division, and not using the correct inverse matrix or vector for division. It is important to double-check the steps and follow the rules carefully to avoid making these mistakes.

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