Matrix multiplication is an important operation in linear algebra. The product of two matrices A and B is defined only if the number of columns of A is equal to the number of rows of B. If A is an m x n matrix and B is an n x p matrix, then their product AB is an m x p matrix.


The basic matrix multiplication method involves multiplying the entries of each row of the first matrix with the corresponding entries in each column of the second matrix and adding the products. This process is repeated for all rows and columns to obtain the entries of the product matrix.


Scalar multiplication is the operation of multiplying a matrix by a scalar, which is a single number. Each entry of the matrix is multiplied by the scalar to obtain the corresponding entry of the resulting matrix.


To find the product A₁A₂A₃A₄ using basic matrix multiplication method, we need to multiply A₁ with A₂, then multiply the result with A₃ and finally multiply the result with A₄. Let's calculate the number of scalar multiplications required for each step.

- A₁A₂: The result will be a 10 x 20 matrix. To find each entry of the resulting matrix, we need to perform 5 scalar multiplications and 4 additions. Therefore, the total number of scalar multiplications required is 10 x 20 x 5 = 1000.
- (A₁A₂)A₃: The result will be a 10 x 10 matrix. To find each entry of the resulting matrix, we need to perform 20 scalar multiplications and 9 additions. Therefore, the total number of scalar multiplications required is 10 x 10 x 20 = 2000.
- ((A₁A₂)A₃)A₄: The result will be a 10 x 5 matrix. To find each entry of the resulting matrix, we need to perform 10 scalar multiplications and 4 additions. Therefore, the total number of scalar multiplications required is 10 x 5 x 10 = 500.

The total number of scalar multiplications required to find the product A₁A₂A₃A₄ using basic matrix multiplication method is 1000 + 2000 + 500 = 3500. Therefore, the correct option is (D) 3500.

C is the answer .