Question

fill in the blank so that the resulting statement is true. the matrix \\(\

$$\begin{bmatrix} 1 & -1 & -4 & \\bigm| & 4 \\\\ 0 & 1 & -18 & \\bigm| & -2 \\\\ 0 & 0 & 0 & \\bigm| & 0 \\end{bmatrix}$$

\\) has ______. the matrix \\(\

$$\begin{bmatrix} 1 & -1 & -4 & \\bigm| & 4 \\\\ 0 & 1 & -18 & \\bigm| & -2 \\\\ 0 & 0 & 0 & \\bigm| & 0 \\end{bmatrix}$$

\\) has \\(\boldsymbol{\bigtriangledown}\\)

Explanation:

Step1: Identify matrix type

The matrix is an augmented matrix (3 rows, 4 columns: 3 variables, 3 equations).

Step2: Check row-echelon form

Leading 1s are in (1,1), (2,2); row 3 is all zeros.

Step3: Analyze consistency

Last row: 0=0 (consistent). Number of non-zero rows (2) < number of variables (3).

Answer:

infinitely many solutions