GATE 2021 BT – Question 54
The determinant of $$A=\begin{pmatrix}1&1&1&1\\-1&1&1&1\\-1&-1&1&1\\1&1&1&3\end{pmatrix}$$ is _______.
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Correct answer: 8
Explanation
**Step 1: simplify with a row operation.** Subtract Row 1 from Row 4 ($R_4\to R_4-R_1$), which does not change the determinant:
$$\begin{pmatrix}1&1&1&1\\-1&1&1&1\\-1&-1&1&1\\0&0&0&2\end{pmatrix}.$$
**Step 2: expand along the last row.** Only the last entry (2) is non-zero, and its cofactor sign is $(+)$ because the position is (4,4):
$$\det A=2\times\det\begin{pmatrix}1&1&1\\-1&1&1\\-1&-1&1\end{pmatrix}.$$
**Step 3: the $3\times3$ determinant.**
$$1(1\cdot1-1\cdot(-1))-1((-1)(1)-(1)(-1))+1((-1)(-1)-(1)(-1))=1(2)-1(0)+1(2)=4 .$$
$$\det A=2\times4=\mathbf{8}.$$