Invertible Matrices

Let%20A%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D2%265%260%5C%5C%200%261%261%5C%5C%20-1%260%263%5Cend%7Bmatrix%7D%5Cright%5D

%5Cleft%7CA%5Cright%7C%3D2%5Cleft(3-0%5Cright)-5%5Cleft(0%2B1%5Cright)

%3D%206%20-%205%20%3D%201

%5Ctherefore%5C%2C%20%5Cleft%7CA%5Cright%7C%3D1

adj%20%5C%2C%20A%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D3%26-1%261%5C%5C%20-15%266%26-5%5C%5C%205%26-2%262%5Cend%7Bmatrix%7D%5Cright%5D%5E%7BT%7D%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D3%26-15%265%5C%5C%20-1%266%26-2%5C%5C%201%26-5%262%5Cend%7Bmatrix%7D%5Cright%5D

A%5E%7B-1%7D%3D%5Cfrac%7Badj%5C%2C%20A%7D%7B%5Cleft%7CA%5Cright%7C%7D%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D3%26-15%265%5C%5C%20-1%266%26-2%5C%5C%201%26-5%262%5Cend%7Bmatrix%7D%5Cright%5D

We are given that $A$ and $B$ are invertible matrices.
We need to determine which of the following statements is not correct:

(A) $%5Ctext%7Badj%7D(A)%20%3D%20%7CA%7C%20A%5E%7B-1%7D$
This is a correct property of adjoint matrices. It holds true that the adjoint of $A$ is equal to $%7CA%7C$ multiplied by the inverse of $A$ .

(B) $%5Cdet(A%5E%7B-1%7D)%20%3D%20%5B%5Cdet(A)%5D%5E%7B-1%7D$
This is also a correct property of determinants. The determinant of the inverse of $A$ is indeed the reciprocal of the determinant of $A$ .

(C) $(AB)%5E%7B-1%7D%20%3D%20B%5E%7B-1%7D%20A%5E%7B-1%7D$
This is a correct property of the inverse of the product of matrices. The inverse of the product $AB$ is equal to the product of the inverses $B%5E%7B-1%7D%20A%5E%7B-1%7D$ , but in reverse order.

(D) $(A%20%2B%20B)%5E%7B-1%7D%20%3D%20B%5E%7B-1%7D%20%2B%20A%5E%7B-1%7D$
This statement is incorrect. The inverse of the sum of two matrices $(A%20%2B%20B)%5E%7B-1%7D$ is not equal to the sum of the individual inverses $B%5E%7B-1%7D%20%2B%20A%5E%7B-1%7D$ . This is a common misconception, and it does not hold for matrix operations.

The correct answer is (D).

High-Yield Trend

Chapter Questions
4 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Invertible Matrices

High-Yield Trend

Chapter Questions 4 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Chapter Questions
4 MCQs