To notice, Every term in a determinant uses each row and column once!

The Pivot Formula

$detP\cdot detA=detL\cdot detU$ gives $detA=+/-(d_1d_2...d_n)$

We assume no row exchanges then $A = LU$ and $A_k = L_kU_k$. Dividing one determinant by the previous determinant ( $det A_k$ divided by $det A_{k-1}$) cancels everything but the latest pivot $d_k$. Each pivot is a ratio of determinants:

The Big Formula for Determinants

Notice the pattern. Each product like $a_{11}a_{23}a_{32}$ has one entry from each row .It also has one entry from each column. The column order 1,3, 2 means that this particular term comes with a minus sign. The column order 3, 1, 2 in $a_{13}a_{21}a_{32}$ has a plus sign (boldface). It will be "permutations" that tell us the sign.

We pay attention only when the entries $a_{ij}$ come from different columns, like (3, 1, 2)

Determinant by Cofactors

When you separate out the factor $a_{11}$ or $a_{12}$ or $a_{1a}$, that comes from the first row, you can see linearity

The cofactors along row 1 are $C_{1j} = ( -1) ^{1+j} det M_{1j}$ . The cofactor expansion is $detA = a_{11}C_{11} + a_{12}C_{12} + · · · + a_{1n}C_{1n}$

Cofactors are useful when matrices have many zeros.