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.

Short answer

The provided statement will never be true.

Step-by-step solution

Step 1- Consider any values of m and n

Consider m as 1 and n as 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.