Question

How to find the determinant of the matrix here ?

Multiple options can be correct.

Answer 1

(A) and (D)

Explanation:

`adj(P)` `=` `((1,4,4),(2,1,7),(1,1,3))`

Edit:
#|(color(red)1, color(blue)4, color(green)4), (color(green)2 ,color(red)1, color(blue)7), (color(blue)1, color(green)1, color(red)3)|#

We multiply the same colors and add them and then we subtract when we multiply the same colors but from this scheme:

`|(color(brown)1, color(orange)4, 4),(color(orange)2, 1, color(brown)7), (1, color(brown)1, color(orange)3)|`

So we have:
`color(red)3+color(blue)28+color(green)8-4-color(orange)24-color(brown)7=4`

Now, since this is the adjoint matrix, and the matrix is 3x3, there is a formula:

`det(adj(A))=(det(A))^(n−1)`

where n is the number of rows in the matrix.

So, we will have
`4=det(A)^2`
`sqrt4=sqrt(det(A)^2)`
`2=|det(A)|`.........................{NOTE that `sqrt(x^2)=|x|`}
`:. det(A) = +-2`

So the answer is A and D.

I hope this helped you.