Q1. - Algebra 2

Question

Determine whether the following statement is always, sometimes, or never true. Explain your reasoning.

 For any matrix$A_{m\times n}$ for$m\ne n$ , $A^2$ is defined.

Step-by-Step Solution

Answer

The provided statement will never be true.

Step 1- Consider any values of m and n

Consider m as 1 and n as 2. 

Step 2 – Write the matrices with dimensions

Split the matrix . $A^2$

$A_{1\times 2}\cdot A_{1\times 2}$

 As the inner dimensions are not the same which means that the matrix$A^2$ will not be defined.

Step 3- Write the conclusion

The provided statement will never be true due to same matrix and different count of row and columns.