Question 1:
Let A be a 4*4 matrix. If determinant of A equal to 10. Then determinant of -2A is
(a) 160
(b) -160
(c) 100
(d) -100
Solution:
\[\left | A \right |=10\]
\[\left | -2A \right |=(-2)^4\times10\]
\[=160\]
Question 2:
Find determinant of the matrix\[\begin{bmatrix} 1 &0 &0 &0 &0 &2 \\ 0 &1 &0 &0 &2 &0 \\ 0 & 0 &1 &2 &0 &0 \\ 0 & 0 & 2 & 1 & 0 & 0\\ 0 & 2 & 0 & 0 & 1 &0 \\ 2 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}\]
(a) 0
(b) -3
(c) -27
(d) 1
Solution:
We know that row transformation does not change the determinant of matrix
\[R_4\rightarrow R_4-2R_3\\R_5\rightarrow R_5-2R_2\\R_6\rightarrow R_6-2R_1\]
\[\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 2\\ 0 & 1 &0 &0 &2 &0 \\ 0 & 0 & 1 & 2 & 0 & 0\\ 0 & 0 & 0 & -3 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 &0 \\ 0&0 &0 &0 &0 &-3 \end{bmatrix}\]
We obtain a upper triangular matrix. Then the determinant is just the product of diagonal entries
So here, the product is\[1\times1\times1\times-3\times-3\times-3=-27\]
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