Question

If A is a 3x3 matrix and detA= -2 then value of detA(adjA) is ?

Answer 1

Answer:

As we know,

|adj A| = |A|^(n-1)

where n is the order of the matrix

Here n = 3

|adj A| = (-2)^(3-1) = (-2)^2 = 4

|adj A| = 4

Now as per your question,

|A|(adj A) = (-2)×(4) = -8

Answer 2

Correct Question :

If A is a 3×3 matrix and |A|= -2 then

value of |adj A| is ?

Theory :

If every element of a square matrix A be replaced by its co-factor , then the Transpose of the matrix so obtained is called the adjoint of matrix A and it is denoted by adj A .

⇒ Properties of Adjoint of a matrix

If A , B are square matrices of order n and $I_{n}$ ,then

1) A(adj A) = |A|$I_{n}$=(adj A)A

2) (adj A) = (adj A)( adj B)

3) |adj A| = |A|${}^{n-1}$

4) adj(adj A) =|A|${}^{n-1}$A

Solution :

Given : |A| = -2

and order of matrix is ,n= 3×3

|adj A| = |A|${}^{n-1}$

⇒ |adj A| = |-2|${}^{3-1}$

⇒ |adj A| = $(-2){}^{2}$

⇒ |adj A| = 4

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More about Adjoint of a matrix :

1) A is singular ⇒ |adj A| = 0

2) A is a diagonals ⇒ adj A is a diagonal

3) A is symmetric⇒ adj A is also symmetric