Mathematical Thinking - Matrices

In summary, matrices are rectangular arrays of numbers or symbols used to represent and manipulate mathematical data and equations. They are useful in mathematical thinking for organizing and solving complex systems of equations and have various real-world applications in fields such as physics, engineering, and computer science. The basic operations that can be performed on matrices include addition, subtraction, multiplication, and division, as well as transposition, inversion, and scalar multiplication. Matrices can also be used to solve equations with multiple variables by setting up a system of equations in matrix form and using mathematical operations. Some common real-world applications of matrices include computer graphics, data compression, optimization problems, machine learning algorithms, financial modeling, and population studies.
  • #1
yarnlife
1
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Hi guys, iv been stuck on this problem for a while now and can't seem to make any headway

construct a counterexample to the following statement:

"for matrices A with real entries 'A^3=Identity impies A=Identity'

im not restricted by size for the matrix.

any hints would help because i just can't think of any counterexamples!

thankyou
 
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  • #2
You may consider $\begin{pmatrix}\cos\varphi&-\sin\varphi\\\sin\varphi&\cos\varphi\end{pmatrix}$, which is the matrix of rotation on the plane by angle $\varphi$ around the origin.
 

FAQ: Mathematical Thinking - Matrices

What are matrices?

Matrices are rectangular arrays of numbers or symbols arranged in rows and columns. They are used to represent and manipulate mathematical data and equations.

How are matrices useful in mathematical thinking?

Matrices are useful in mathematical thinking because they provide a way to organize and solve complex systems of equations. They are also used in various fields such as physics, engineering, and computer science to solve real-world problems.

What are the basic operations that can be performed on matrices?

The basic operations that can be performed on matrices include addition, subtraction, multiplication, and division. Matrices can also be transposed, inverted, and multiplied by a scalar (a single number).

Can matrices be used to solve equations with multiple variables?

Yes, matrices can be used to solve equations with multiple variables. This is done by setting up a system of equations in matrix form and then using mathematical operations to solve for the variables.

Are there any real-world applications of matrices?

Yes, matrices have many real-world applications, such as computer graphics, data compression, and optimization problems. They are also used in machine learning algorithms, financial modeling, and population studies.

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