Question

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

17: Rank$\left|\begin{array}{ccc}2& 2& 2\\ 2& 2& 2\\ 2& 2& 2\end{array}\right|\mathbf{=}\mathbf{2}$

Short answer

False, Rank of $\left|\begin{array}{ccc}2& 2& 2\\ 2& 2& 2\\ 2& 2& 2\end{array}\right|=1$

Step-by-step solution

Step 1:Rank of matrix 

For finding the rank of matrix first of all we will change the given matrix in the echelon form.

We have given the matrix.

$\left|\begin{array}{ccc}2& 2& 2\\ 2& 2& 2\\ 2& 2& 2\end{array}\right|$

Justification of answer  

Now, reduce the given matrix into row echelon form.

$$\begin{gathered} \left|\begin{array}{ccc}2& 2& 2\\ 2& 2& 2\\ 2& 2& 2\end{array}\right| \\ {R}_2\to R_2-R_1 \\ {R}_3\to R_3-R_1 \\ ~\left|\begin{array}{ccc}2& 2& 2\\ 2& 2& 2\\ 2& 2& 2\end{array}\right| \end{gathered}$$

Now, the given matrix is in echelon form. Thus, the number of non-zero rows is equal to the rank of the matrix.

Thus, the rank of the given matrix is 1.