GATE CSE 2000 | Question: 1.3

Question

The determinant of the matrix $$\begin{bmatrix}2 &0  &0  &0 \\  8&  1&  7& 2\\  2&  0&2  &0 \\  9&0  & 6 & 1 \end{bmatrix}$$

• A. $4$
• B. $0$
• C. $15$
• D. $20$

Comments

Ans

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how long does this method work  (of Best Answer)  and when it doesn't ?

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@jeetsdo you have the solution to your problem now?

Answer 1

$\text{Let }|A|=\begin{vmatrix}2 &0  &0  &0 \\  8&  1&  7& 2\\  2&  0&2  &0 \\  9&0  & 6 & 1 \end{vmatrix}$

$\implies |A|=2\times\begin{vmatrix}  1&  7& 2\\  0&2  &0 \\ 0  & 6 & 1 \end{vmatrix}$

$\implies |A|=2\times 1\times \begin{vmatrix}  2  &0 \\ 6 & 1 \end{vmatrix}$

$\implies |A|=2\times 1\times (2-0)$

$\implies |A|=2\times 1\times 2=4$

Answer$: A$

Comments

@ Soumya29  the determinant you calculated as product of diagonal entries is after applying elimination method, right?

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@meghna..  No..I applied it directly and this is a general procedure. You can apply it to find the determinant of any matrix that has lots of 0's in it. Upper triangular one is a special case of this method only. You can check the reference link in my above for it.

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reference link not working

 

Answer 2

I reduced it to an upper triangular matrix.

Comments

@Pronomita Dey 1 

after reducing R4=R4-9/2R1 , 

in row 4th  column  3rd  there should be 6.

you wrote 0???

Answer 3

2*1*2*1 = 4 so option A is right

Answer 4

2*1*2*1=4 break it into determinant of smaller pieces where you are doing it in such a way that you should have to do lesser number of operations

Comments

if we do nornally by cofactor method we get 20 as ans why that's not true?

Answer 5

in this type of  question ask what is determinant of 4x4 matrix then first check which row or column has maximum number of zero’s.then find  co-factor of this row and it will answer.

for example  first i will take first row and first element

co-factor=2*(2*1-0)=4.

Answer 6

Answer is 4