Question

A matrix P of order 2 × 3 with each entry 0 or 1 and α is a scalar which is 3 or 4. If R = αA, the number of matrices R formed is :

• A. 63
• B. 64
• C. 128
• D. 127

Answer 1

The Correct Option is B

To solve this problem, we need to determine the number of matrices R, given that R = αP, where P is a matrix with each entry either 0 or 1, and α is a scalar which is either 3 or 4.

First, consider the matrix P, which is of order 2 × 3, meaning it has 2 rows and 3 columns, hence a total of 6 elements. Each element in P can independently be 0 or 1. This gives:

Number of possible P matrices = 2⁶ = 64.

For each matrix P, scalar multiplication by α = 3 or 4 will give a unique matrix R. Each element in P, which is either 0 or 1, when multiplied by α, remains either 0 (if the element was 0) or will be α (if the element was 1). Therefore, the multiplication does not change the number of distinct matrices that can be formed. Hence, changing α will not affect the count of the unique transformations from matrix P to matrix R.

Therefore, the total number of matrices R that can be formed is equal exactly to the number of matrices P.

Thus, the number of matrices R formed is 64.