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).

About Invertible Matrices - KCET

Invertible Matrices is a vital chapter for KCET aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Invertible Matrices PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Invertible Matrices carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
4 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

About Invertible Matrices - KCET

Frequently Asked Questions

Why focus on Invertible Matrices PYQs?

How to best use this analysis?

Invertible Matrices

High-Yield Trend

Chapter Questions 4 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

About Invertible Matrices - KCET

Frequently Asked Questions

Why focus on Invertible Matrices PYQs?

How to best use this analysis?

Chapter Questions
4 MCQs