Question

Find the rank of the matrices in 2 through 4.

3.$\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$

Short answer

The rank of the matrix$\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$ is, $1$.

Step-by-step solution

Find row reduce echelon form 

The number of leading 1’s in$\mathbf{rref}\mathbf{\left(}\mathbf{A}\mathbf{\right)}$represents the rank of a matrix A denoted by$\mathbf{rank}\mathbf{\left(}\mathbf{A}\mathbf{\right)}$.

The given matrix is, $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$.

The reduced row echelon form of the given matrix is:

$rref\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

Determine the rank of a matrix. 

The number of leading 1’s in the matrix $\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$are$1$.

Hence, the rank of the given matrix is 1.