JEE MAIN - Mathematics (2017 - 8th April Morning Slot - No. 3)
Let A be any 3 $$ \times $$ 3 invertible matrix. Then which one of the following is not always true ?
adj (A) = $$\left| \right.$$A$$\left| \right.$$.A$$-$$1
adj (adj(A)) = $$\left| \right.$$A$$\left| \right.$$.A
adj (adj(A)) = $$\left| \right.$$A$$\left| \right.$$2.(adj(A))$$-$$1
adj (adj(A)) = $$\left| \, \right.$$A $$\left| \, \right.$$.(adj(A))$$-$$1
Explanation
We know, the formula
A-1 = $${{adj\left( A \right)} \over {\left| A \right|}}$$
$$ \therefore $$ adj (A) = $$\left| \right.$$A$$\left| \right.$$.A$$-$$1
So, Option (A) is true.
We know, the formula
adj (adj (A)) = $${\left| A \right|^{n - 2}}.A$$
Now if we put n = 3 as given that A is a 3 $$ \times $$ 3 matrix, we get
adj (adj (A)) = $${\left| A \right|^{3 - 2}}.A$$ = $$\left| A \right|.A$$
So, Option (B) is also true.
We know, the formula
adj (adj (A)) = $${\left| A \right|^{n - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$
Now if we put n = 3 as given that A is a 3 $$ \times $$ 3 matrix, we get
adj (adj (A)) = $${\left| A \right|^{3 - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$ = $${\left| A \right|^{2}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$
So, Option (C) is also true.
Now in this formula
adj (adj (A)) = $${\left| A \right|^{n - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$
if we put n = 2, we get
adj (adj (A)) = $${\left| A \right|^{2 - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$ = $${\left| A \right|}{\left( {adj\left( A \right)} \right)^{ - 1}}$$
But as A is a 3 $$ \times $$ 3 matrix so we can not take n = 2, so we can say for a 3 $$ \times $$ 3 matrix option (D) is not true.
So, Option (D) is false.
A-1 = $${{adj\left( A \right)} \over {\left| A \right|}}$$
$$ \therefore $$ adj (A) = $$\left| \right.$$A$$\left| \right.$$.A$$-$$1
So, Option (A) is true.
We know, the formula
adj (adj (A)) = $${\left| A \right|^{n - 2}}.A$$
Now if we put n = 3 as given that A is a 3 $$ \times $$ 3 matrix, we get
adj (adj (A)) = $${\left| A \right|^{3 - 2}}.A$$ = $$\left| A \right|.A$$
So, Option (B) is also true.
We know, the formula
adj (adj (A)) = $${\left| A \right|^{n - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$
Now if we put n = 3 as given that A is a 3 $$ \times $$ 3 matrix, we get
adj (adj (A)) = $${\left| A \right|^{3 - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$ = $${\left| A \right|^{2}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$
So, Option (C) is also true.
Now in this formula
adj (adj (A)) = $${\left| A \right|^{n - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$
if we put n = 2, we get
adj (adj (A)) = $${\left| A \right|^{2 - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$ = $${\left| A \right|}{\left( {adj\left( A \right)} \right)^{ - 1}}$$
But as A is a 3 $$ \times $$ 3 matrix so we can not take n = 2, so we can say for a 3 $$ \times $$ 3 matrix option (D) is not true.
So, Option (D) is false.
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