$$
\bar{A}=\left(\begin{array}{llll|l}
1 & 2 & 3 & 4 & 1 \
0 & 5 & 6 & 7 & 2 \
0 & 0 & 8 & 9 & 3 \
0 & 0 & 0 & 0 & a
\end{array}\right)
$$
a. unique solution
b. infinite number of solutions
c. no solutions

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We start by analyzing the given augmented matrix: $$ \bar{A}=\left(\begin{array}{llll|l} 1 & 2 & 3 & 4 & 1 \ 0 & 5 & 6 & 7 & 2 \ 0 & 0 & 8 & 9 & 3 \ 0 & 0 & 0 & 0 & a \end{array}\right) $$ ‖ ‖ step 2 ⋮ Notice that the last row of the matrix is $[0 \; 0 \; 0 \; 0 \; a]$. If $a = 0$, the last row becomes $[0 \; 0 \; 0 \; 0 \; 0]$, which indicates a row of zeros in the augmented matrix. ‖ ‖ step 3 ⋮ A row of zeros in the augmented matrix implies that the system of equations is consistent if the other rows do not lead to a contradiction. ‖ ‖ step 4 ⋮ Since the other rows do not lead to a contradiction and the matrix is in row echelon form, the system has infinitely many solutions. ‖ ***B*** ∻Key Concept∻ ⚹Row of zeros in augmented matrix⚹ ∻Explanation∻ ⚹A row of zeros in the augmented matrix indicates that the system is consistent and has infinitely many solutions if there are no contradictions in the other rows.⚹◊What is the determinant of the matrix A in terms of the variable a⍭ Generate me a similar question◊

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