Give example of matrices such that AB=AC but B=/=C

In summary, an example of matrices A, B, and C in M(2,R) such that AB=AC, but B is not equal to C is A = 1 0 0 0, B = 0 0 0 0, and C = 0 1 1 0.
  • #1
xlalcciax
12
0
1. Let M(2,R) be the set of all 2 x 2 matrics over R. Give an example of matrices A,B,C in M(2,R) such that AB=AC, but B is not equal to C.


3.
 
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  • #2


xlalcciax said:
1. Let M(2,R) be the set of all 2 x 2 matrics over R. Give an example of matrices A,B,C in M(2,R) such that AB=AC, but B is not equal to C.


3.
What have you tried? You have to show some effort before we can provide any help.
 
  • #3


Mark44 said:
What have you tried? You have to show some effort before we can provide any help.

(A^-1)AB=(A^-1)AC so B=C. This shows that A must have no inverse element. So A could be
1 0
0 0
because det(a)=1-0=0 so A has no inverse. I don't know what A and B could be.
 
  • #4


xlalcciax said:
(A^-1)AB=(A^-1)AC so B=C. This shows that A must have no inverse element.
No, what this shows is that if A is invertible (has an inverse), then AB = AC implies that B = C. But you're not given that A is invertible.
xlalcciax said:
So A could be
1 0
0 0
because det(a)=1-0=0 so A has no inverse. I don't know what A and B could be.
You mean B and C. See if you can cobble up different matrices B and C so that AB = 0 and AC = 0, but B != C.
 
  • #5


Mark44 said:
No, what this shows is that if A is invertible (has an inverse), then AB = AC implies that B = C. But you're not given that A is invertible.

You mean B and C. See if you can cobble up different matrices B and C so that AB = 0 and AC = 0, but B != C.

what does AB = 0 mean? does it mean det(A) x det(B) or matrix A x matrix B?
 
  • #6


By 0 I meant the 2 x 2 matrix whose entries are all 0.
 
  • #7


Mark44 said:
By 0 I meant the 2 x 2 matrix whose entries are all 0.

so they could be B = 0 0 and C = 0 0 ??
...... 0 1....1 0
 
  • #8


Sure, why not? All you had to do was come up with three 2 x 2 matrices such that AB = AC, but B != C. It looks like you did just what you are asked to do.
 

1. What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is commonly used in mathematical operations and represents a set of equations or linear transformations.

2. What does AB=AC mean in terms of matrices?

AB=AC means that the product of matrix A and B is equal to the product of matrix A and C. In other words, the two matrices result in the same value when multiplied by matrix A.

3. Can you give an example of matrices where AB=AC but B ≠ C?

Yes, for example:

A = [1 2; 3 4] and B = [2 -1; 4 -2] and C = [4 -2; 8 -4]

When multiplied by matrix A, both B and C result in the same value of [2 3; 4 5], but B is not equal to C.

4. What is the significance of AB=AC but B ≠ C?

This shows that matrix multiplication is not commutative, meaning the order of multiplication matters. While AB=AC, B and C are not interchangeable in the equation.

5. How can this concept be applied in real-life situations?

Matrices are used in various fields such as physics, economics, and computer science to model and solve complex systems. For example, in economic analysis, matrices can represent supply and demand relationships, and understanding their properties can help predict market outcomes.

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