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Chapter 3: Problem 15

Row reduce each of the following matrices to echelon form: (a) \(A=\left[\begin{array}{cccc}1 & 2 & -3 & 0 \\ 2 & 4 & -2 & 2 \\ 3 & 6 & -4 & 3\end{array}\right]\) (b) \(B=\left[\begin{array}{rrr}-4 & 1 & -6 \\ 1 & 2 & -5 \\ 6 & 3 & -4\end{array}\right]\)

Short Answer

Expert verified
The echelon forms of the given matrices are: (a) \(A = \left[\begin{array}{cccc}1 & 2 & -3 & 0 \\\ 0 & 0 & 1 & 0.5 \\\ 0 & 0 & 0 & 0.5\end{array}\right]\) (b) \(B = \left[\begin{array}{rrr}1 & 0 & \frac{1}{9} \\\ 0 & 1 & -\frac{26}{9} \\\ 0 & 0 & 0\end{array}\right]\)

Step by step solution

01

Eliminate below the leading entry (1)

Subtract 2 times row 1 from row 2 and also subtract 3 times row 1 from row 3: \[\left[\begin{array}{cccc}1 & 2 & -3 & 0 \\\ 0 & 0 & 4 & 2 \\\ 0 & 0 & 5 & 3\end{array}\right]\]
02

Working with row 2

Since the leading entry in row 2 is 4, divide row 2 by 4 to make it 1: \[\left[\begin{array}{cccc}1 & 2 & -3 & 0 \\\ 0 & 0 & 1 & 0.5 \\\ 0 & 0 & 5 & 3\end{array}\right]\]
03

Eliminate below the new leading entry

To eliminate below the new leading entry (1 in row 2), subtract 5 times row 2 from row 3: \[\left[\begin{array}{cccc}1 & 2 & -3 & 0 \\\ 0 & 0 & 1 & 0.5 \\\ 0 & 0 & 0 & 0.5\end{array}\right]\] This is the echelon form of matrix A. (b) For the matrix B: \(B=\left[\begin{array}{rrr}-4 & 1 & -6 \\\ 1 & 2 & -5 \\\ 6 & 3 & -4\end{array}\right]\)
04

Swap rows to have a non-zero leading entry

Swap row 1 and row 2 to obtain a non-zero leading entry. \[\left[\begin{array}{rrr}1 & 2 & -5 \\\ -4 & 1 & -6 \\\ 6 & 3 & -4\end{array}\right]\]
05

Eliminate below the leading entry (1)

Add 4 times row 1 to row 2 and subtract 6 times row 1 from row 3: \[\left[\begin{array}{rrr}1 & 2 & -5 \\\ 0 & 9 & -26 \\\ 0 & -9 & 26\end{array}\right]\]
06

Working with row 2

Since the leading entry in row 2 is 9, divide row 2 by 9 to make it 1: \[\left[\begin{array}{rrr}1 & 2 & -5 \\\ 0 & 1 & -\frac{26}{9} \\\ 0 & -9 & 26\end{array}\right]\]
07

Eliminate below and above the new leading entry

Add 9 times row 2 to row 3 and subtract 2 times row 2 from row 1: \[\left[\begin{array}{rrr}1 & 0 & \frac{1}{9} \\\ 0 & 1 & -\frac{26}{9} \\\ 0 & 0 & 0\end{array}\right]\] This is the echelon form of matrix B.

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