Definition of Row Reduced Matrix:

A row reduced matrix (or reduced row echelon form, RREF) is a matrix that meets the following conditions:

1. Leading 1s: In each row, the first non-zero element is a 1 (called a leading 1).
2. Zeros above and below leading 1s: The column containing a leading 1 has all other elements in that column equal to zero.
3. Row ordering: Rows with all zero entries, if they exist, are at the bottom of the matrix.
4. Left-to-right progression: The leading 1 in each row is to the right of the leading 1 in the row above it.

Given Matrices:

Matrix A:

1 & 3 & 2 \\ 2 & 6 & 4 \\ -1 & -3 & 0 \end{pmatrix}$$ Matrix B: $$B = \begin{pmatrix} 2 & 1 & 7 \\ 4 & 5 & 8 \\ 6 & 3 & 9 \end{pmatrix}$$ ### Steps to row reduce Matrix A and Matrix B: Let me perform the row reduction for both matrices. #### Row reduction of Matrix A: I'll row-reduce matrix $$A$$ first. $$A = \begin{pmatrix} 1 & 3 & 2 \\ 2 & 6 & 4 \\ -1 & -3 & 0 \end{pmatrix}$$ Now, for Matrix $$B$$, I'll row reduce it as well. #### Row reduction of Matrix B: $$B = \begin{pmatrix} 2 & 1 & 7 \\ 4 & 5 & 8 \\ 6 & 3 & 9 \end{pmatrix}$$ Let me calculate both reductions.### Row Reduced Form of Matrices A and B: For matrix $$A$$: $$A = \begin{pmatrix} 1 & 3 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$$ For matrix $$B$$: $$B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ ### Interpretation: - The row-reduced form of matrix $$A$$ has two leading 1s and a row of zeros at the bottom. - Matrix $$B$$ is reduced to the identity matrix, which is its row-reduced form. Would you like further details on the row reduction process? ### Related Questions: 1. What is the difference between row echelon form and reduced row echelon form? 2. How can row reduction be applied to solve systems of linear equations? 3. Can every matrix be reduced to a unique reduced row echelon form? 4. How does Gaussian elimination differ from Gauss-Jordan elimination? 5. How can we determine the rank of a matrix from its row-reduced form? ### Tip: Row reduction can be used to find the rank of a matrix, which is the number of non-zero rows in its row echelon form.