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,
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,
(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,
Updated On: Nov 17, 2025
- only (I) is TRUE
only (II) is TRUE
- both (I) and (II) are TRUE
- neither (I) nor (II) is TRUE
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The Correct Option is C
Solution and Explanation
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:
-
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.
-
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
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