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Section 3.7 Elementary Matrices

Definition 3.7.1.

An elementary matrix is a matrix that when multiplied on the left of a given matrix performs one of the 3 elementary row operations for reducing a matrix.

Find \(3 \times 3\) elementary matrices \(\mtx{E}\) that do the following actions. For each, also find det(\(\mtx{E}\)) and \(\mtx{E}^{-1}\text{.}\)

  1. switch rows 1 and 3

  2. multiply row 3 by the number 5

  3. add 10 times row 2 to row 3

Compute determinants for the following matrices. Also find the elementary matrices indicated and their determinant. Choose carefully how to compute each determinant.

  1. \(\mtx{A}\) = \(\left[ \begin{array}{rrr}-1\amp 2\amp 3\\4\amp 5\amp 6\\7\amp 8\amp 9 \end{array} \right]\)

  2. \(\mtx{E}_1\mtx{A} = \left[ \begin{array}{rrr}4\amp 5\amp 6\\-1\amp 2\amp 3\\7\amp 8\amp 9 \end{array} \right]\) where \(\mtx{E}_1 =\)

  3. \(\mtx{E}_2\mtx{A} = \left[ \begin{array}{rrr}-10\amp 20\amp 30\\4\amp 5\amp 6\\7\amp 8\amp 9 \end{array} \right]\) where \(\mtx{E}_2 =\)

  4. \(\displaystyle \mtx{B} = \left[ \begin{array}{rrr}-1\amp 2\amp 3\\-1\amp 2\amp 3\\7\amp 8\amp 9 \end{array} \right]\)

  5. \(\mtx{E}_3\mtx{A} = \left[ \begin{array}{ccc}-1\amp 2\amp 3\\4+-1\amp 5+2\amp 6+3\\7\amp 8\amp 9 \end{array} \right]\) where \(\mtx{E}_3 =\)

  6. \(\displaystyle \mtx{C} = \left[ \begin{array}{rrr}-1\amp 2\amp 3\\4\amp 5\amp 6\\0\amp 0\amp 0 \end{array} \right]\)

  7. \(\mtx{A}^T = \left[ \begin{array}{rrr}-1\amp 4\amp 7\\2\amp 5\amp 8\\3\amp 6\amp 9 \end{array} \right]\) (This is called the transpose of \(\mtx{A}\))

(Elementary Matrix Lemma) For any \(n \times n\) matrix \(\mtx{M}\) and any \(n \times n\) elementary matrix \(\mtx{E}\text{,}\) find a relationship between det(\(\mtx{EM}\)), det(\(\mtx{E}\)) and det(\(\mtx{M}\)) .