### Matrix calculus: Determinants

### Row and column reduction

Let #A# be a square matrix. In order to achieve a situation where *expansion along a row or column* is useful for calculating the determinant of #A#, we employ *row reduction*. This allows us to produce many zeros in the matrix while the determinant changes in a controlled way. As we have seen *before*, an *elementary row operation* on a matrix #A# may be seen as multiplication of #A# from the left by a square matrix, so the *product formula for the determinant* can be used. Thus, some bookkeeping is needed to record the determinants of matrices used while performing the row reduction. As we will see, since #\det(A) = \det(A^\top)#, column operations work just as well.

Elementary operations in terms of matrices The effect of elementary row operations on a matrix #A# can be described as follows in terms of matrix multiplication.

- \(R_i\rightarrow \lambda \cdot R_i\;(\lambda \neq 0)\): Multiplication of row #i# of #A# by a number #\lambda # is equivalent to multiplication from the left by the diagonal matrix #D_{i,\lambda }# with #(i,i)#-entry equal to #\lambda # and all other entries on the diagonal equal to #1#.
- \(R_i\rightarrow R_i+\lambda \cdot R_j\;(i\ne j)\): Addition of the multiple #\lambda # of one row #j# to another row #i# is equivalent to multiplication from the left by the matrix #E_{ij,\lambda }#, whose #(i,j)#-entry is equal to #\lambda#, whose diagonal elements are equal to #1#, and all of whose other elements are equal to #0#.
- \(R_i\leftrightarrow R_j\;(i\ne j)\): Interchange of the rows #i# and #j# of #A# is equivalent to multiplying from the left by the
*permutation matrix*#P_{(i,j)}# associated with the*transposition*#(i,j)#.

We apply this interpretation of row and column operations to compute determinants.

Effect of elementary operations on the determinant The effect of elementary row or column operations on the determinant of a square matrix #A# is indicated in the table below. If #B# is the matrix from the second column, then the determinant of the result #B\,A# or #A\,B# of the operation is equal to #\det(B)\cdot \det(A)#.

elementary operation | matrix | determinant |

\(R_i\rightarrow \lambda \cdot R_i\;(\lambda \neq 0)\) | \(D_{i,\lambda }\) | \(\lambda \) |

\(R_i\rightarrow R_i+\lambda \cdot R_j\;(i\ne j)\) | \(E_{ij,\lambda }\) | \(1\) |

\(R_i\leftrightarrow R_j\;(i\ne j)\) | \(P_{(i,j)}\) | \(-1\) |

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