Let \(\mathrm{A}=\left[\begin{array}{ll}\mathrm{a} & \mathrm{b} \\ \mathrm{c} & \mathrm{d}\end{array}\right]\) and \(\mathrm{B}=\left[\begin{array}{l}\alpha \\ \beta\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0\end{array}\right]\) such that \(\mathrm{AB}=\mathrm{B}\) and \(\mathrm{a}+\mathrm{d}=2021\), then the value of \(\mathrm{ad}-\mathrm{bc}\) is equal to ______.

JEE Main · 2021 · 17 Mar 2021 · Shift 2 · Matrices

Let

A = [a c b d]

and

B = [α β] \neq = [00]

such that

AB = B

and

a + d = 2021

, then the value of

ad - bc

is equal to ______.

Answer: 2020

The relation

A B = B

can be written as

(A - I) B = O

, a homogeneous system in

α

and

β

. Since

B

is not the zero column, the system has a non-trivial solution, which forces the coefficient matrix to be singular:

det (A - I) = a - 1 c b d - 1 = 0

Expanding gives

(a - 1) (d - 1) - b c = 0

, that is

a d - b c - (a + d) + 1 = 0

. With

a + d = 2021

a d - b c = (a + d) - 1 = 2021 - 1

a d - b c = 2020

JEE Main · 2021 · 17 Mar 2021 · Shift 2 · Integer answer