Compute the determinant of the 4x4 matrix with rows (19,5,7,8), (18,4,4,4), (9,1,2,1) and (9,2,3,4).

Step-by-step solution

1. 1

   Use row 3 to attack the first column. Row $3$ begins with $9$, and the other first-column entries are $19$, $18$ and $9$. Subtracting multiples of row $3$ (an operation that preserves the determinant) gives

   $R_1\leftarrow R_1-2R_3,\quad R_2\leftarrow R_2-2R_3,\quad R_4\leftarrow R_4-R_3$

   $\begin{vmatrix}1&3&3&6\\0&2&0&2\\9&1&2&1\\0&1&1&3\end{vmatrix}$

   Note $19-18=1$, so the first column is not fully cleared — one more step is needed.

2. 2

   Finish clearing the column with the new leading 1.

   $R_3\leftarrow R_3-9R_1\quad\Longrightarrow\quad (0,\ 1-27,\ 2-27,\ 1-54)=(0,-26,-25,-53)$

   $\begin{vmatrix}1&3&3&6\\0&2&0&2\\0&-26&-25&-53\\0&1&1&3\end{vmatrix}$

3. 3

   Expand along the first column. Only the leading $1$ remains, with cofactor sign $+$:

   $\det=1\cdot\begin{vmatrix}2&0&2\\-26&-25&-53\\1&1&3\end{vmatrix}$

4. 4

   Simplify and evaluate the 3×3. Factor $2$ out of the first row:

   $2\begin{vmatrix}1&0&1\\-26&-25&-53\\1&1&3\end{vmatrix}=2\Big[1(-75+53)-0+1(-26+25)\Big]=2\big[-22-1\big]=2(-23)=-46$

   $\boxed{-46}$

5. 5

   Cross-check the size against the sister problem. The very similar matrix with rows $(18,5,7,8),(18,3,4,4),(9,1,1,1),(9,2,3,4)$ has determinant $9$; changing four entries here has moved it to $-46$, including a sign flip. Because determinants are alternating multilinear functions, no smooth relationship between the two values should be expected — each has to be computed independently.