PX153 - K6 - properties of determinants

$(i) :$

det A = det A^{T}

$(i i) :$

det A B = det A det B

$(i i i) :$

det I_{n} = 1

$(i v) :$

det λ A = λ^{n} det A

$(v) :$ if only one row is multiplied by $λ$ , $det A$ is multiplied by $λ$
$(v i) :$ if $B$ is obtained by interchanging two rows/columns of a matrix, $A$ , then, $det B = - det A$
$(v i i) :$ if $r o w (i)$ is replaced by $[r o w (i) + λ \times r o w (j)]$ , where $i \neq j$ , the determinant is unchanged
$(v i i i) :$ if two rows are identical, the determinant is zero
eg:

A = [\begin{matrix} 1 & i & - 1 \\ 0 & - 1 & - i \\ 1 & i & 0 \end{matrix}]

- expanding the $1^{s t}$ row: $det A = - 1$
- expanding the $3^{r d}$ row: $det A = - 1$

notes for examples about reducing $4 \times 4$ matrix to easily calculate the determinant