Q.Let $\Delta = \begin{vmatrix} a & p & x \\ b & q & y \\ c & r & z \end{vmatrix} = 16$, then $\Delta_1 = \begin{vmatrix} p + x & a + x & a + p \\ q + y & b + y & b + q \\ r + z & c + z & c + r \end{vmatrix} = 32$.

Name the columns of $\Delta$: $A=(a,b,c)^T$, $P=(p,q,r)^T$, $X=(x,y,z)^T$, so $\Delta=\det[A,P,X]=16$.

The columns of $\Delta_1$ are $P+X,\ A+X,\ A+P$. Expanding by column-linearity gives $2^3=8$ determinants; every one with a repeated column is $0$, leaving

$\Delta_1 = \det[P,X,A] + \det[X,A,P].$ …

Set up

Let the columns of the original determinant be

The columns of $\Delta_1$ are $P+X$, $A+X$, $A+P$, so $\Delta_1 = \det[P+X,\ A+X,\ A+P]$.

Expand by multilinearity

A determinant is linear in each column separately, so splitting all three columns produces $2\times2\times2 = 8$ determinants whose columns are chosen from $\{P,X\}$, $\{A,X\}$, $\{A,P\}$. Any determinant with two equal columns is $0$. Discarding those, the only surviving choices are

$\Delta_1 = \det[P,X,A] + \det[X,A,P].$ …

Method: Evaluating a Determinant Built From Permuted Column-Sums of a Known Determinant

Use this method whenever a new determinant's columns are sums/combinations of the columns of an original determinant $\Delta$ whose value you already know.

Steps

Step 1: Name the original columns as vectors

If $\Delta=\det[\mathbf A,\mathbf P,\mathbf X]$ is given, write the new determinant's columns in terms of $\mathbf A,\mathbf P,\mathbf X$.

Step 2: Expand every summed column by multilinearity

Splitting each column that is a sum produces $2^{(\text{number of split columns})}$ determinants (Step 2 of the counting method above), each with columns chosen from the original vectors.

Step 3: Discard every determinant with a repeated column

Any determinant where two columns coincide (e.g. both chosen as $\mathbf A$) is automatically $0$ — cross those out immediately; only genuinely distinct-column determinants survive.

Step 4: Identify each surviving determinant as a permutation of $\Delta$ …

Common Mistakes

Mistake 1: Forgetting to discard determinants with a repeated column after splitting

Why it's wrong: expanding all $8$ split-off determinants without removing the ones with two identical columns leads to extra, incorrect terms in the sum. Correct approach: after splitting, cross out every determinant whose columns aren't all distinct — only $2$ of the $8$ survive here.

Mistake 2: Misjudging whether a column permutation is even or odd …