Use elementary row operations to reduce the given matrix to...

use elementary row operations to reduce the given matrix to row-echelon form, and hence determine the rank of each matrix.
$$\left[\begin{array}{lll}
0 & 1 & 3 \\
0 & 1 & 4 \\
0 & 3 & 5
\end{array}\right]$$.

Key Concepts

Elementary Row Operations

Elementary row operations are the manipulations that can be performed on the rows of a matrix to simplify it. These include swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another row. These operations are fundamental because they maintain the solution set of the associated linear system and are used in algorithms like Gaussian elimination.

Row-Echelon Form

A matrix is in row-echelon form if it has a staircase pattern where all nonzero rows are above any rows of all zeros, and each leading entry (pivot) of a row is strictly to the right of the leading entry of the row above it. This form is crucial in solving systems of linear equations, as it simplifies back-substitution and develops an understanding of the structure of the matrix.

Pivot Position

Pivot positions are the locations in a matrix that correspond to the leading entries in each row once the matrix is transformed into row-echelon form. Identifying these positions helps in determining the rank of the matrix and analyzing the linear independence of the rows or columns.

Matrix Rank

The rank of a matrix is the number of pivot positions in its row-echelon form. It indicates the maximum number of linearly independent rows or columns in the matrix, which is key to understanding the solution space of the accompanying system of linear equations and assessing the matrix's invertibility.

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