Question

What are some examples of equivalent matrices?

Answer 1

We have a question asking the examples of equivalent matrices. We all know that, in order for the matrices to be equivalent they need to satisfy some mathematical conditions, and only then can they be termed as equivalent matrices.
The conditions that we will be looking for in the two matrices will be:
The matrices have same dimension
The matrices have the same order.
The corresponding elements of the matrices are equal.
In easier words, two matrices are said to be equal if they have the same number of rows and columns for admissible values.

Complete step-by-step solution:
Now, since we have been asked about equivalent matrices, we need to have two matrices to compare in the first place.
Here we have our first matrix,
A = \left[ {\begin{array}{*{20}{c}} 2&{ - 3}&{ - 1} \\\ 4&0&{ - 5} \\\ 6&3&7 \end{array}} \right];
Also, we have our second matrix,
B = \left[ {\begin{array}{*{20}{c}} 2&{ - 3}&{ - 1} \\\ 4&0&{ - 5} \\\ 6&3&7 \end{array}} \right];
Now, for further more explanation, we have a third matrix which has the values,
C = \left[ {\begin{array}{*{20}{c}} 2 \\\ { - 9} \end{array}} \right];
Now, we have matrix $A,B,C$ and we can clearly see that the number of rows and columns don’t match in the matrix $A$ and matrix $C$. This is one way of understanding how matrices are equivalent or not.
Some other examples of equivalent matrices are,
A = \left[ {\begin{array}{*{20}{c}} 1&3 \\\ 2&4 \end{array}} \right]and B = \left[ {\begin{array}{*{20}{c}} 1&3 \\\ 2&4 \end{array}} \right]

Note: Matrix equivalence is an equivalence relation on the space of rectangular matrices. If we have two matrices that are of the same value then the following conditions are also checked like, whether they have the same rank. Having the same rank is one of the most important conditions for having equivalent matrices. Also, it’s also checked whether the matrices can be transformed into one another by a combination of rows and columns operation.