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Found in: Page 34

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Find the rank of the matrices in 2 through 4.4. $\left[\begin{array}{ccc}1& 4& 7\\ 2& 5& 8\\ 3& 6& 9\end{array}\right]$

The rank of the matrix$\left[\begin{array}{ccc}1& 4& 7\\ 2& 5& 8\\ 3& 6& 9\end{array}\right]$ is, $2$.

See the step by step solution

## Step 1: 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, $\left[\begin{array}{ccc}1& 4& 7\\ 2& 5& 8\\ 3& 6& 9\end{array}\right]$ .

The reduced row echelon form of the given matrix is:

$rref\left[\begin{array}{ccc}1& 4& 7\\ 2& 5& 8\\ 3& 6& 9\end{array}\right]=\left[\begin{array}{ccc}1& 0& -1\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$

## Step 2: Determine the rank of a matrix.

The number of leading 1’s in the matrix $\left[\begin{array}{ccc}1& 0& -1\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$ are $2$.

Hence, the rank of the given matrix is 2.