Linear Algebra question concerning matrices

In summary, the conversation discusses the question of whether, given three matrices A, B, and C where AB=AC and A is not equal to zero, it necessarily follows that B=C. The individual discussing the topic looks at the associative laws of matrix multiplication and concludes that B and C would be the same, but is advised to check the law again and consider the cases when C=0 and the matrices are not commutative. The summary also clarifies the use of the term "rank" in this context.
  • #1
insertnamehere
50
0
Hi, I have a question about matrices.
If A,B,C are matrices such that AB=AC and A is not equal to zero, does it follow that B=C?
I looked at the associative laws that A(BC)=(AB)C=B(AC), and I think that B and C would be the same. Am I on the right track?
 
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  • #2
I'd check that associative law again. The first two work, but because you can't multiply matrices together unless their ranks match up correctly, the B(AC) will not work for all cases.

For the AB=AC... you're on the right track. Try multiplying by a matrix and see if you can prove it for yourself. Another hint is below in white if you need it

You'll need to do something to A and premultiply
 
  • #3
enigma said:
I'd check that associative law again. The first two work, but because you can't multiply matrices together unless their ranks match up correctly, the B(AC) will not work for all cases.

Of course you mean the numbers of rows and columns of the matrices are such that the multiplication B(AC) might not be defined even if A(BC) is. "Rank" is a bad word here as it has another meaning (I'm being picky, but thought it was worth clarifying).

Another problem with insertnamehere's associativity law is swapping the order of multiplication of matrices is not allowed in general (multiplication is not commutative).

As to AB=AC implying B=C, associativitity has nothing to do with it. Try considering the case when C=0. Then your proposition says "if A is nonzero and AB=0 then B is non zero". Is this always true? (it might help to think about 2x2 examples here)
 

1. What are the basic operations for matrices?

The basic operations for matrices include addition, subtraction, and multiplication. Addition and subtraction can be performed on matrices of the same size, while multiplication can be performed on matrices of compatible sizes.

2. How do you determine the size of a matrix?

The size of a matrix is determined by the number of rows and columns it has. For example, a matrix with 3 rows and 4 columns would be considered a 3x4 matrix. The size of a matrix is always written as "number of rows x number of columns".

3. What is 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. The inverse of a matrix can only exist if the original matrix is square and has a non-zero determinant.

4. How do you solve a system of linear equations using matrices?

To solve a system of linear equations using matrices, you can use the process of elimination. First, write the equations in matrix form, with the variables as the columns and the constants as the last column. Then, use elementary row operations to reduce the matrix to row-echelon form. Finally, use back substitution to find the values of the variables.

5. What is the significance of determinants in linear algebra?

Determinants are used to determine if a matrix is invertible and to find the inverse of a matrix. They also have applications in solving systems of linear equations and calculating the area or volume of a parallelogram or parallelepiped defined by the matrix. In addition, determinants play a role in eigenvalue and eigenvector calculations.

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