Nonzero Matrcies A; B; C such that AC = BC and A does not equal B?

• sheldonrocks97
In summary, to complete the given problem, you can use the simplest non-zero matrices (shown above) and play around with them to find two non-zero matrices A, B, and C, such that AC = BC and A does not equal B. This can be achieved by using the property that if (A-B)C = 0, then AC - BC = 0. You can also explore the concept of singular matrices to help you in this problem.
sheldonrocks97
Gold Member

Homework Statement

Find nonzero matrices A; B; C such that AC = BC
and A does not equal B

The Attempt at a Solution

I'm not sure where to start, I would like to know how to complete this problem.

sheldonrocks97 said:

Homework Statement

Find nonzero matrices A; B; C such that AC = BC
and A does not equal B

The Attempt at a Solution

I'm not sure where to start, I would like to know how to complete this problem.

The simplest non-zero matrices are
$$\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},\quad \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},\quad \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},\quad \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$$
Play around.

If AC = BC, then AC - BC = 0. I.e. (A-B)C = 0.

Can you find two non-zero matrices that multiply to zero?

1. What is a nonzero matrix?

A nonzero matrix is a matrix that contains at least one nonzero element. This means that it is not a matrix full of zeros.

2. What does it mean for two matrices to be equal?

Two matrices are considered equal if they have the same dimensions and all corresponding elements are equal. In other words, they have the same number of rows and columns, and each element in the same position in both matrices has the same value.

3. Can two matrices have the same product but be different?

Yes, it is possible for two matrices A and B to have the same product AB, but for A and B to be different matrices. This is because matrix multiplication depends on the dimensions of the matrices, not just their values.

4. Is it possible for AC = BC if A is not equal to B?

Yes, it is possible for two matrices A and B to be different, but for their products with another matrix C to be equal. This can happen if the matrices have the same number of columns, but different number of rows.

5. How can nonzero matrices with equal products be useful in mathematics?

Nonzero matrices with equal products are useful for solving equations and systems of equations. By using matrix multiplication, we can find values for the matrices that satisfy the equation or system, even if the matrices themselves are not equal.

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