GO Classes DPP | GATE CS | Linear Algebra | Determinant

Answer 1

Apply row operations to convert into upper triangular form.

Swap $R_1 \leftrightarrow R_6$,
$R_2 \leftrightarrow R_5$,
$R_3 \leftrightarrow R_4$

Total swaps $= 3$

So determinant gets multiplied by $(-1)^3 = -1$

New matrix becomes:

$\begin{vmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
0 & 2 & 3 & 4 & 5 & 6 \\
0 & 0 & 3 & 4 & 5 & 6 \\
0 & 0 & 0 & 4 & 5 & 6 \\
0 & 0 & 0 & 0 & 5 & 6 \\
0 & 0 & 0 & 0 & 0 & 6
\end{vmatrix}$

Now matrix is upper triangular.

Determinant is product of diagonal elements:

$1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 = 720$

Including sign:

$\det = -720$

Correct Option: $B$

Answer 2

-720

Reverse the matrix and multiply all diagonal 6! and add - sign. Because we are reversing

Answer 3

2 Methods to solve these. a) Upper triangular property b) Block method

Lets solve using Upper triangular property

Step 1: Convert into upper triangular by swaping the rows i.e R1 with R6, R2 with R5, R3 with R4, we will acheive an upper triangular matrix.

Step 2 :By using upper triangular property, the multiplication of diagonal = determinant of the matrix

Step 3: = 1 x 2 x 3 x 4 x 5 x 6

           = 720

Step 4: As 3 swaps were done, -1 x -1 x -1 = -1

Step 5: 720 x -1 = -720

Therefore option B)