Question

$\left[\begin{array}{ccc}3& -1& 2\\ -3& 1& 2\\ -6& 2& 4\end{array}\right]$What is the rank of the matrix.

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is B

2

The rank of a matrix is simply the maximum number of linearly independent vectors in a matrix. And this number is equal to the number of non-zero rows in its row echelon matrix. Therefore, to find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows. So first step would be to convert the given matrix into Row echelon form.

$R_2\to R_2+R_1$ $R_3\to R_3+2R_1$

$\left[\begin{array}{ccc}3& -1& 2\\ 0& 0& 4\\ 0& 0& 8\end{array}\right]$

$R_1\to R_1\frac{1}{3}$, $R_2\to R_2\left(\frac{1}{4}\right)$, $R_3\to R_3\left(\frac{1}{8}\right)$

$\left[\begin{array}{ccc}3& \frac{-1}{3}& \frac{2}{3}\\ 0& 0& 1\\ 0& 0& 1\end{array}\right]$

$R_3\to R_3–R_2$

$\left[\begin{array}{ccc}1& \frac{-1}{3}& \frac{2}{3}\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]$

$\therefore$ we have 2 non zero rows $\to$ Rank = 2.