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. Determine the rank of the matrix:
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Q. . Determine the rank of the matrix:
- Write Matrix : Write down the matrix .We need to determine the rank of this matrix, which is the dimension of the vector space spanned by its rows (or columns).
- Check Row Dependence: Check if any rows are linearly dependent.For matrix , no rows are multiples of each other, so we cannot immediately identify any linearly dependent rows.
- Perform Row Reduction: Perform row reduction to echelon form to identify the rank.We will use elementary row operations to convert the matrix into row echelon form.Starting with matrix :A = \left[\begin{array}{ccc}\(\newline1 & 1 & 1 (\newline\)0 & 1 & 1 (\newline\)1 & 0 & 1\end{array}\right]\)Subtract the first row from the third row to make the first element of the third row zero:A = \left[\begin{array}{ccc}\(\newline1 & 1 & 1 (\newline\)0 & 1 & 1 (\newline\)0 & -1 & 0\end{array}\right]\)Now, add the third row to the second row to make the second element of the second row zero:A = \left[\begin{array}{ccc}\(\newline1 & 1 & 1 (\newline\)0 & 0 & 1 (\newline\)0 & -1 & 0\end{array}\right]\)Finally, add the third row to the first row to make the second element of the first row zero:A = \left[\begin{array}{ccc}\(\newline1 & 0 & 1 (\newline\)0 & 0 & 1 (\newline\)0 & -1 & 0\end{array}\right]\)
- Identify Non-Zero Rows: Identify the number of non-zero rows in the row echelon form.In the row echelon form of matrix , we have:A = \left[\begin{array}{ccc}\(\newline1 & 0 & 1 (\newline\)0 & 0 & 1 (\newline\)0 & -1 & 0\end{array}\right]\)All rows are non-zero, which means they are linearly independent.
- Determine Rank: Determine the rank of the matrix.Since all three rows are linearly independent, the rank of matrix is equal to the number of non-zero rows in its row echelon form.
