Question

Find the rank of the matrix

$\left[\begin{array}{ccc}a& b& c\\ 0& d& e\\ 0& 0& f\end{array}\right]$,

where a, d and f are nonzero, and b,c and e are arbitrary numbers.

Short answer

The rank of the matrix $\left[\begin{array}{ccc}a& b& c\\ 0& d& e\\ 0& 0& f\end{array}\right]$is, 3.

Step-by-step solution

Consider the systems

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

Consider the matrix,

$\left[\begin{array}{ccc}a& b& c\\ 0& d& e\\ 0& 0& f\end{array}\right]$

Determine the rank of a matrix

The matrix $\left[\begin{array}{ccc}a& b& c\\ 0& d& e\\ 0& 0& f\end{array}\right]$,is already in its reduced row-echelon form with the upper triangular matrix.

As $a,d,f$are non-zero, can be considered as 1’s.

Hence, 3 is the rank of the matrix$\left[\begin{array}{ccc}a& b& c\\ 0& d& e\\ 0& 0& f\end{array}\right]$.