Question

For which values of a, b, c, d, and eis the following matrix in reduced row-echelon form?

$\mathbf{A}\mathbf{=}\left[\begin{array}{l}1ab30-2\\ 00c1d3\\ 0e0011\end{array}\right]$

Short answer

Values if given arbitrary value for the matrix to be in reduced echelon form are

$\mathrm{a}=2,\mathrm{e}=1,\mathrm{c}=1,\mathrm{d}=0,\mathrm{b}=0$.

Step-by-step solution

Step 1: Finding the values

For finding the value of a, b, c, d, e in the given matrix we will follow the following steps:

For the value of a:

$$\begin{gathered} A\mathbf{=}\mathbf{}\left[\begin{array}{c}1ab30-2\\ 00c1d3\\ 0e0011\end{array}\right] \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{} \end{gathered}$$

Since, 1^st column has 1 which is our 1^st pivot entry.

Step 2: Finding the value of e and a

Since, e and a is in the second column which not has any pivot entry.

Hence, e and a can have any value. Let the value of e be 1and value of a be 2.

Step 3: Finding the value of c and b

Since, c is in the third column which is at position of 2^nd pivot entry.

Hence, c must be equal to 1.

Since, b is in the third column which is above the c. Since c is pivot entry then b must be 0.

Step 4: Finding the value of d

Since, d is in the fifth column which has our 3^rd pivot entry. Hence, d must be 0.

Step 5: Final answer

Values if given arbitrary value for the matrix to be in reduced echelon form are

$\mathrm{a}=2,\mathrm{e}=1,\mathrm{c}=1,\mathrm{d}=0,\mathrm{b}=0$

Matrix will be

$\mathbf{A}\mathbf{=}\left[\begin{array}{c}12030-2\\ 001103\\ 010011\end{array}\right]$