Find all real numbers a, b and c such that the following matrix is in reduced row echelon form.
⎛
⎜
⎝
a 1 0 1
0 0 1 b
0 0 c 0
⎞
⎟
⎠

Question

Find all real numbers a, b and c such that the following matrix is in reduced row echelon form.
⎛
⎜
⎝
a 1 0 1
0 0 1 b
0 0 c 0
⎞
⎟
⎠

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the values of $a$, $b$, and $c$ such that the given matrix is in reduced row echelon form (RREF), we need to ensure that the matrix satisfies the conditions for RREF. ‖ ‖ step 2 ⋮ The given matrix is: $$ \begin{pmatrix} a & 1 & 0 & 1 \ 0 & 0 & 1 & b \ 0 & 0 & c & 0 \ \end{pmatrix} $$ ‖ ‖ step 3 ⋮ In RREF, each leading entry in a row must be 1, and all entries in the column containing a leading 1 must be zero. ‖ ‖ step 4 ⋮ For the second row, the leading entry is already 1 in the third column. Therefore, $b$ can be any real number. ‖ ‖ step 5 ⋮ For the third row, the leading entry must be 1. Therefore, $c$ must be 1. ‖ ‖ step 6 ⋮ For the first row, the leading entry must be 1. Therefore, $a$ must be 0. ‖ ‖ step 7 ⋮ The values of $a$, $b$, and $c$ that make the matrix in RREF are: $$ a = 0, \quad b = \text{any real number}, \quad c = 1 $$ ‖ ∻Answer∻ ⚹ $a = 0$, $b$ can be any real number, $c = 1$ ⚹ ∻Key Concept∻ ⚹Reduced Row Echelon Form (RREF)⚹ ∻Explanation∻ ⚹In RREF, each leading entry in a row must be 1, and all entries in the column containing a leading 1 must be zero. This determines the values of $a$, $b$, and $c$.⚹◊What is the pivoting process in row echelon form?⍭ Generate me a similar question◊

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