Question

The determinant of matrix \( A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ -1 & 1 & 1 & 1 \\ -1 & -1 & 1 & 1 \\ 1 & 1 & 1 & 3 \end{pmatrix} \) is \(\underline{\hspace{1cm}}\). 

Hint:

Row-reduction to triangular form is the fastest way to compute determinants of 4×4 matrices.

Answer 1

Correct Answer: 8

Using determinant expansion or row operations, simplify the matrix.
Performing: \(R_1 \rightarrow R_1 - R_2\), \(R_2 \rightarrow R_2 - R_3\), \(R_4 \rightarrow R_4 - R_1\).
The matrix reduces to an upper-triangular form with diagonal entries whose product equals 8.
Thus, \[ \det(A) = 8 \]