Question

Determine whether the statements that follow are true or false, and justify your answer.

47.If $\mathit{a}\mathit{d}\mathbf{-}\mathit{b}\mathit{c}\mathbf{\ne }\mathbf{0}$, then the matrix$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ must have rank 2.

Short answer

The statement, $ad-bc\ne 0$“If , then the matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$must have rank 2 .” is false.

Step-by-step solution

Consider the condition

Consider the matrix.

$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$

The reduced row-echelon form of the matrix is,

$\left(\begin{array}{cc}a& b\\ 0& d-\frac{bc}{a}\end{array}\right)$

Thus, the condition is, $a\ne 0$.

Conclusion

Hence, for the matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ to have a rank of 2 , the condition to be satisfied is, $a\ne b$.