Invertible Matrices

The number of 3\times3 matrices whose entries are either 0 or 1 is calculated as follows: each entry has 2 choices (either 0 or 1). Therefore, the total number of such matrices is 2 9 = 512. Next, we need to compute how many of these matrices have a sum of all entries that is a prime number. The sum of the entries is a number between 0 and 9 (inclusive) since the matrix can have 0 ones (all zero matrix) to 9 ones (all one matrix). We evaluate which of these sums are prime numbers. The prime numbers between 0 and 9 are: 2, 3, 5, 7. We compute the number of matrices for each prime sum: Sum = 2: Choose 2 positions from 9 to be 1, i.e., combinations of 9 taken 2 = C(9,2)=36. Sum = 3: Choose 3 positions from 9 to be 1, i.e., combinations of 9 taken 3 = C(9,3)=84. Sum = 5: Choose 5 positions from 9 to be 1, i.e., combinations of 9 taken 5 = C(9,5)=126. Sum = 7: Choose 7 positions from 9 to be 1, i.e., combinations of 9 taken 7 = C(9,7)=36. Adding these gives: 36 + 84 + 126 + 36 = 282 This confirms that the computed number of such matrices is 282, which is within the expected range (282,282). Therefore, the number of 3\times3 matrices with entries 0 or 1, and a prime sum is 282 .

To determine which matrix cannot be obtained from the given matrix $%5Cbegin%7Bbmatrix%7D%20-1%20%26%202%20%5C%5C%201%20%26%20-1%20%5Cend%7Bbmatrix%7D$ by a single elementary row operation, let's first recall the three types of elementary row operations:

Swapping two rows.
Multiplying a row by a non-zero scalar.
Adding or subtracting a multiple of one row to another row.

Let's examine the options one by one:

$%5Cbegin%7Bbmatrix%7D%200%20%26%201%20%5C%5C%201%20%26%20-1%20%5Cend%7Bbmatrix%7D$

To achieve this, we can add the first row to the second row:

R_1%20%2B%20R_2%20%5Crightarrow%20R_1

%5Cbegin%7Bbmatrix%7D%20-1%20%26%202%20%5C%5C%201%20%26%20-1%20%5Cend%7Bbmatrix%7D%20%5Crightarrow%20%5Cbegin%7Bbmatrix%7D%200%20%26%201%20%5C%5C%201%20%26%20-1%20%5Cend%7Bbmatrix%7D

$%5Cbegin%7Bbmatrix%7D%201%20%26%20-1%20%5C%5C%20-1%20%26%202%20%5Cend%7Bbmatrix%7D$

To obtain this matrix, multiply the first row by $-1$ and the second row by $-1$ :

-R_1%20%5Crightarrow%20R_1

and

-R_2%20%5Crightarrow%20R_2

%5Cbegin%7Bbmatrix%7D%20-1%20%26%202%20%5C%5C%201%20%26%20-1%20%5Cend%7Bbmatrix%7D%20%5Crightarrow%20%5Cbegin%7Bbmatrix%7D%201%20%26%20-1%20%5C%5C%20-1%20%26%202%20%5Cend%7Bbmatrix%7D

$%5Cbegin%7Bbmatrix%7D%20-1%20%26%202%20%5C%5C%20-2%20%26%207%20%5Cend%7Bbmatrix%7D$

To achieve this transformation, it requires adding some scalar multiple of the first row to the second row; let's verify:

%5Cbegin%7Bbmatrix%7D%20-1%20%26%202%20%5C%5C%201%20%26%20-1%20%5Cend%7Bbmatrix%7D%20%5Crightarrow%20%5Cbegin%7Bbmatrix%7D%20-1%20%26%202%20%5C%5C%20(-2)(-1)%20%2B%201%20%26%20(-2)(2)%20%2B%20(-1)%20%5Cend%7Bbmatrix%7D

%5CRightarrow%20(-2)%20%5Ctimes%20R_1%20%2B%20R_2%20%5Crightarrow%20R_2

By calculating: $-2%20%5Ctimes%20(-1)%20%2B%201%20%3D%202%20%2B%201%20%3D%203$ and $-2%20%5Ctimes%202%20%2B%20(-1)%20%3D%20-4%20-%201%20%3D%20-5$ . Thus incorrect, it cannot be obtained with a single operation.

$%5Cbegin%7Bbmatrix%7D%20-1%20%26%202%20%5C%5C%20-1%20%26%203%20%5Cend%7Bbmatrix%7D$

Achieve this by subtracting the first row from the second row:

R_2%20-%20R_1%20%5Crightarrow%20R_2

%5Cbegin%7Bbmatrix%7D%20-1%20%26%202%20%5C%5C%201%20%26%20-1%20%5Cend%7Bbmatrix%7D%20%5Crightarrow%20%5Cbegin%7Bbmatrix%7D%20-1%20%26%202%20%5C%5C%20-1%20%26%203%20%5Cend%7Bbmatrix%7D

Therefore, the matrix that cannot be obtained from the original matrix $%5Cbegin%7Bbmatrix%7D%20-1%20%26%202%20%5C%5C%201%20%26%20-1%20%5Cend%7Bbmatrix%7D$ by a single elementary row operation is $%5Cbegin%7Bbmatrix%7D%20-1%20%26%202%20%5C%5C%20-2%20%26%207%20%5Cend%7Bbmatrix%7D$ .

About Invertible Matrices - JEE-MAIN

Invertible Matrices is a vital chapter for JEE-MAIN 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
2 MCQs

Official Solution

Official Solution

About Invertible Matrices - JEE-MAIN

Frequently Asked Questions

Why focus on Invertible Matrices PYQs?

How to best use this analysis?

Invertible Matrices

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

About Invertible Matrices - JEE-MAIN

Frequently Asked Questions

Why focus on Invertible Matrices PYQs?

How to best use this analysis?

Chapter Questions
2 MCQs