Question

Find the rank of each of the following matrices.

$\left(\begin{array}{ccc}1& 1& 2\\ 2& 4& 6\\ 3& 2& 5\end{array}\right)$

Short answer

The rank of the matrix$\left(\begin{array}{ccc}1& 1& 2\\ 2& 4& 6\\ 3& 2& 5\end{array}\right)$is 2 .

Step-by-step solution

Given information

The given matrix is$\left(\begin{array}{ccc}1& 1& 2\\ 2& 4& 6\\ 3& 2& 5\end{array}\right)$

Definition of the rank of a matrix

The number of non-zero rows remaining in a row reduced matrix is called the rank of a matrix.

Apply row operations to reduce the matrix into row reduction form

Consider the matrix A =$\left(\begin{array}{ccc}1& 1& 2\\ 2& 4& 6\\ 3& 2& 5\end{array}\right)$

Multiply row 1 by and subtract the result from row 2.

$A=\left(\begin{array}{ccc}1& 1& 2\\ 0& 2& 2\\ 3& 2& 5\end{array}\right)$ $R_2-2R_1$

Multiply row 1 by $-3$and subtract the result from row 3.

$A=\left(\begin{array}{ccc}1& 1& 2\\ 0& 2& 2\\ 0& -1& -1\end{array}\right)$ role="math" localid="1653389234639" $R_3-3R_1$

Divide the second row by 2.

$A=\left(\begin{array}{ccc}1& 1& 2\\ 0& 1& 1\\ 0& -1& -1\end{array}\right)\frac{1}{2}{R}_2$

Add row 2 to row 3.

$A=\left(\begin{array}{ccc}1& 1& 2\\ 0& 1& 1\\ 0& 0& 0\end{array}\right)R_3\pm R_2$

The row reduced form of the given matrix is $\left(\begin{array}{ccc}1& 1& 2\\ 0& 1& 1\\ 0& 0& 0\end{array}\right)$

Here, the total number of non-zero rows is 2. Hence, the rank of the matrix

$A=\left(\begin{array}{ccc}1& 1& 2\\ 2& 4& 6\\ 3& 2& 5\end{array}\right)is2$