Question

Let 𝑀=𝑀₁𝑀₂, where 𝑀₁ and 𝑀₂ are two 3×3 distinct matrices. Consider the following two statements:
(I) The rows of 𝑀 are linear combinations of rows of 𝑀₂. 
(II) The columns of 𝑀 are linear combinations of columns of 𝑀₁. 
Then,

• A. only (I) is TRUE
• B. only (II) is TRUE
• C. both (I) and (II) are TRUE
• D. neither (I) nor (II) is TRUE

Answer 1

The Correct Option is C

To solve the problem, we need to analyze the properties of the matrix multiplication \(M = M_1M_2\). Here, \(M_1\) and \(M_2\) are distinct 3x3 matrices. Let's examine the given statements separately:

1. Statement I: The rows of \(M\) are linear combinations of the rows of \(M_2\).
   • Matrix multiplication is defined such that the resultant matrix's rows are formed from the linear combinations of the rows of the first matrix and the elements of the second matrix.
   • In \(M = M_1M_2\), the result matrix \(M\) inherits properties from the multiplication process.
   • Here, the rows of \(M\) are linear combinations of the rows of \(M_1\).
   • Therefore, Statement I is FALSE.
2. Statement II: The columns of \(M\) are linear combinations of the columns of \(M_1\).
   • During matrix multiplication, the columns of the resultant matrix are linear combinations of the columns of the second matrix utilized in the multiplication process.
   • Thus, the columns of \(M\) are indeed linear combinations of the columns of \(M_1\).
   • Therefore, Statement II is TRUE.

Based on the analysis, the correct conclusion is that both statements I and II hold true within the context of the problem. As a result, the correct answer is:

• both (I) and (II) are TRUE