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Statement-1 (Assertion and Statement- 2 ...

Statement-1 (Assertion and Statement- 2 (Reason)
Each of these questions also has four alternative
choices, only one of which is the correct answer. You
have to select the correct choice as given below.
Statement-1 For a singular matrix `A , if AB = AC rArr B = C`
Statement-2 If `abs(A) = 0,` thhen` A^(-1)` does not exist.

A

Statement- is true, Statement -2 is true, Statement-2
is a correct explanation for Statement-1

B

Statement-1 is true, Statement-2 is true, Sttatement - 2
is not a correct explanation for Stamtement-1

C

Statement 1 is true, Statement - 2 is false

D

Statement-1 is false, Statement-2 is true

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze both statements provided in the question. ### Step 1: Analyze Statement 1 **Statement 1**: For a singular matrix \( A \), if \( AB = AC \) then \( B = C \). - A singular matrix is defined as a matrix that does not have an inverse, which occurs when its determinant is zero, i.e., \( \text{det}(A) = 0 \). - The equation \( AB = AC \) can be rearranged as \( AB - AC = 0 \) or \( A(B - C) = 0 \). - Since \( A \) is singular, we cannot conclude that \( B - C = 0 \) (or \( B = C \)) because multiplying by a singular matrix can yield the zero matrix without \( B \) and \( C \) being equal. Thus, **Statement 1 is false**. ### Step 2: Analyze Statement 2 **Statement 2**: If \( \text{det}(A) = 0 \), then \( A^{-1} \) does not exist. - By definition, a matrix \( A \) has an inverse \( A^{-1} \) if and only if its determinant is non-zero. - Therefore, if \( \text{det}(A) = 0 \), it directly implies that the matrix \( A \) is singular and hence does not have an inverse. Thus, **Statement 2 is true**. ### Conclusion - **Statement 1** is false. - **Statement 2** is true. The correct choice based on the analysis is that **Statement 1 is false and Statement 2 is true**. ### Final Answer The correct option is **D**: Statement 1 is false, Statement 2 is true. ---
To solve the problem, we need to analyze both statements provided in the question. ### Step 1: Analyze Statement 1 **Statement 1**: For a singular matrix \( A \), if \( AB = AC \) then \( B = C \). - A singular matrix is defined as a matrix that does not have an inverse, which occurs when its determinant is zero, i.e., \( \text{det}(A) = 0 \). - The equation \( AB = AC \) can be rearranged as \( AB - AC = 0 \) or \( A(B - C) = 0 \). - Since \( A \) is singular, we cannot conclude that \( B - C = 0 \) (or \( B = C \)) because multiplying by a singular matrix can yield the zero matrix without \( B \) and \( C \) being equal. ...
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