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Let for any matrix M ,M−1exist. Which of the following is not true.
- a)(M−1)2 = (M2)−1
- b)(M−1)−1 = (M)
- c)(M−1)−1 = (M−1)1
- d)none of these.
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Let for any matrix M ,M−1exist.Which of the following is not tru...
Clearly , (M−1)−1 = (M−1)1 is not true.
Community Answer
Let for any matrix M ,M−1exist.Which of the following is not tru...
Given information: For any matrix M, M1 exists.
We need to find the statement that is not true among the given options.
Explanation:
Let's first understand what M1 represents. M1 can be any matrix obtained by converting the rows of matrix M into columns. Let's say M has n rows and m columns, then M1 will have m rows and n columns.
Using this understanding, let's check each option.
a) (M1)2 = (M2)1
Here, (M1)2 represents the second row of matrix M1 and (M2)1 represents the first column of matrix M2.
Since M1 has m rows and n columns, M2 will have n rows and m columns. Therefore, (M2)1 will have n elements and (M1)2 will have m elements.
Since the number of elements in (M1)2 and (M2)1 are different, this statement is not true. Hence, option a) is not true.
b) (M1)1 = (M)
Here, (M1)1 represents the first row of matrix M1 and (M) represents matrix M.
Since M1 has m rows and n columns, (M1)1 will have n elements and (M) will have n rows and m columns.
We can obtain the first row of matrix M1 by taking the first column of matrix M and converting it into row form. Therefore, (M1)1 = transpose of first column of M.
Since the transpose of a column vector is equal to the row vector, (M1)1 = (M). Hence, option b) is true.
c) (M1)1 = (M1)1
This statement is trivially true as any matrix is always equal to itself. Hence, option c) is also true.
d) None of these
Since options a), b), and c) have been checked, and option c) is true, the correct answer is option c).
Conclusion:
The statement that is not true among the given options is option a).
We need to find the statement that is not true among the given options.
Explanation:
Let's first understand what M1 represents. M1 can be any matrix obtained by converting the rows of matrix M into columns. Let's say M has n rows and m columns, then M1 will have m rows and n columns.
Using this understanding, let's check each option.
a) (M1)2 = (M2)1
Here, (M1)2 represents the second row of matrix M1 and (M2)1 represents the first column of matrix M2.
Since M1 has m rows and n columns, M2 will have n rows and m columns. Therefore, (M2)1 will have n elements and (M1)2 will have m elements.
Since the number of elements in (M1)2 and (M2)1 are different, this statement is not true. Hence, option a) is not true.
b) (M1)1 = (M)
Here, (M1)1 represents the first row of matrix M1 and (M) represents matrix M.
Since M1 has m rows and n columns, (M1)1 will have n elements and (M) will have n rows and m columns.
We can obtain the first row of matrix M1 by taking the first column of matrix M and converting it into row form. Therefore, (M1)1 = transpose of first column of M.
Since the transpose of a column vector is equal to the row vector, (M1)1 = (M). Hence, option b) is true.
c) (M1)1 = (M1)1
This statement is trivially true as any matrix is always equal to itself. Hence, option c) is also true.
d) None of these
Since options a), b), and c) have been checked, and option c) is true, the correct answer is option c).
Conclusion:
The statement that is not true among the given options is option a).
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