Can AB^2009 be calculated?Matrix Multiplication and Exponentiation Limitations

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In summary, the conversation discusses the possibility of computing AB^2009, with the participants considering the size of matrices A and B and the implications of using an exponent. Ultimately, it is determined that it is possible to compute AB^2009 as long as A and B are both square matrices of the same size.
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MattRSK
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Homework Statement



Matrix A and B are both square matrices of the same size.

Is it possible to compute,

AB^2009

If not, why not?

The Attempt at a Solution



Just had this question in an exam, the exam did not allow calculators and I could not answer this question, I gather it is not possible but I cannot explain why.

Because A and B are square matrices it is possible to multiply them together but I am unsure about the exponent. Any ideas?
 
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  • #2
Suppose that A and B are both n x n matrices. Then what is the size of AB? Then what about ABB = AB^2? And ABBB = AB^3? ...
 
  • #3
Well if A and B are both n x n then they will always be n x n matrices. So does that mean that it is possible?
 
  • #4
By the way, is that AB2009[/b] or (AB)2009?

It doesn' really matter. The only important point is that any two n by n matrices can be multiplied and give an n by n matrix as a result- which can be multiplied again by any other n by n matrix.
 

FAQ: Can AB^2009 be calculated?Matrix Multiplication and Exponentiation Limitations

What are matrix operations?

Matrix operations are mathematical operations that involve manipulating matrices, which are rectangular arrays of numbers. Examples of matrix operations include addition, subtraction, multiplication, and inversion.

Why are matrix operations important?

Matrix operations are important because they are used to solve a wide range of problems in various fields such as physics, engineering, economics, and computer graphics. They allow us to represent and manipulate data in a compact and efficient manner.

What is the difference between matrix multiplication and scalar multiplication?

Matrix multiplication is a binary operation that combines two matrices to create a new matrix, while scalar multiplication is a unary operation that multiplies a matrix by a single number. In other words, matrix multiplication involves combining two matrices, while scalar multiplication involves scaling a single matrix.

Can matrices of different sizes be multiplied together?

No, matrices of different sizes cannot be multiplied together. In order for matrix multiplication to be possible, the number of columns in the first matrix must be equal to the number of rows in the second matrix. This is known as the "dimension rule" for matrix multiplication.

How do I know if a matrix is invertible?

A square matrix is invertible if its determinant is non-zero. If the determinant is zero, the matrix is singular and cannot be inverted. Another way to check for invertibility is to perform row reduction on the matrix and see if it reduces to the identity matrix. If it does, then the matrix is invertible.

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