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If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equal to
- a)- (3 A2 + 2 A + 5I)
- b)3 A2 + 2 A + 5I
- c)- 3 A2 - 2 A - 5
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equa...
3A3 + 2A2 + 5A + I = 0
3A3 + 2A2 + 5A + AA−1 = 0
A−1 = −3A2 − 2A − 5
3A3 + 2A2 + 5A + AA−1 = 0
A−1 = −3A2 − 2A − 5
Most Upvoted Answer
If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equa...
Solution:
Given, 3A3 - 2A2 + 5A - I = 0
To find A-1, we need to use the formula:
A-1 = adj(A) / |A|
Where adj(A) is the adjugate of A, and |A| is the determinant of A.
Step 1: Finding the determinant of A
3A3 - 2A2 + 5A - I = 0
Multiplying both sides by A-1, we get
3A2 - 2A + 5I - A-1 = 0
Rearranging the terms, we get
3A2 - (A-1)2 + 5I = 0
Multiplying both sides by (A-1)2, we get
3A2(A-1)2 - (A-1)4 + 5I(A-1)2 = 0
Expanding the terms, we get
3A4 - 8A3 + 7A2 - 2A + 5I = 0
Multiplying both sides by A-1, we get
3A3 - 8A2 + 7A - 2I + 5A-1 = 0
Adding I to both sides, we get
3A3 - 8A2 + 7A + 5A-1 = I
Taking determinant on both sides, we get
|3A3 - 8A2 + 7A + 5A-1| = |I|
|A||3A2 - 8A + 7I + 5A-1| = 1
|A|(3A - 7I)(A - 1I)(A + 5I) = 1
|A| = -1/15
Step 2: Finding the adjugate of A
The adjugate of a matrix A is given by the transpose of its cofactor matrix. The cofactor matrix is obtained by replacing each element of A with its corresponding determinant, and then multiplying it by (-1)i+j, where i and j are the row and column indices of the element.
Using this formula, we get
C11 = |A22 A23 A24| = |-2 -3 -1| = -4
C12 = |-A12 A13 A14| = |-1 2 0| = 2
C13 = |A12 -A13 A14| = |1 -1 1| = 1
C14 = |-A12 -A13 A14| = |-1 -1 0| = 1
C21 = |-A21 A23 A24| = |-2 -2 -1| = -3
C22 = |A11 A13 A14| = |3 0 0| = 3
C23 = |-A11 -A13 A14| = |-3 1 1| = 1
C24 = |A11 -A13 -A14| = |3 -1 -1| = -1
C31 = |A21 A22 A24| = |-1 1 -1| = 1
C32 = |-A11 A12 A14| = |-3 -1
Given, 3A3 - 2A2 + 5A - I = 0
To find A-1, we need to use the formula:
A-1 = adj(A) / |A|
Where adj(A) is the adjugate of A, and |A| is the determinant of A.
Step 1: Finding the determinant of A
3A3 - 2A2 + 5A - I = 0
Multiplying both sides by A-1, we get
3A2 - 2A + 5I - A-1 = 0
Rearranging the terms, we get
3A2 - (A-1)2 + 5I = 0
Multiplying both sides by (A-1)2, we get
3A2(A-1)2 - (A-1)4 + 5I(A-1)2 = 0
Expanding the terms, we get
3A4 - 8A3 + 7A2 - 2A + 5I = 0
Multiplying both sides by A-1, we get
3A3 - 8A2 + 7A - 2I + 5A-1 = 0
Adding I to both sides, we get
3A3 - 8A2 + 7A + 5A-1 = I
Taking determinant on both sides, we get
|3A3 - 8A2 + 7A + 5A-1| = |I|
|A||3A2 - 8A + 7I + 5A-1| = 1
|A|(3A - 7I)(A - 1I)(A + 5I) = 1
|A| = -1/15
Step 2: Finding the adjugate of A
The adjugate of a matrix A is given by the transpose of its cofactor matrix. The cofactor matrix is obtained by replacing each element of A with its corresponding determinant, and then multiplying it by (-1)i+j, where i and j are the row and column indices of the element.
Using this formula, we get
C11 = |A22 A23 A24| = |-2 -3 -1| = -4
C12 = |-A12 A13 A14| = |-1 2 0| = 2
C13 = |A12 -A13 A14| = |1 -1 1| = 1
C14 = |-A12 -A13 A14| = |-1 -1 0| = 1
C21 = |-A21 A23 A24| = |-2 -2 -1| = -3
C22 = |A11 A13 A14| = |3 0 0| = 3
C23 = |-A11 -A13 A14| = |-3 1 1| = 1
C24 = |A11 -A13 -A14| = |3 -1 -1| = -1
C31 = |A21 A22 A24| = |-1 1 -1| = 1
C32 = |-A11 A12 A14| = |-3 -1
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If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equa...
Option A is correct
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If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equal toa)- (3 A2 + 2 A + 5I)b)3 A2 + 2 A + 5Ic)- 3 A2 - 2 A - 5d)none of theseCorrect answer is option 'C'. Can you explain this answer?
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If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equal toa)- (3 A2 + 2 A + 5I)b)3 A2 + 2 A + 5Ic)- 3 A2 - 2 A - 5d)none of theseCorrect answer is option 'C'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equal toa)- (3 A2 + 2 A + 5I)b)3 A2 + 2 A + 5Ic)- 3 A2 - 2 A - 5d)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equal toa)- (3 A2 + 2 A + 5I)b)3 A2 + 2 A + 5Ic)- 3 A2 - 2 A - 5d)none of theseCorrect answer is option 'C'. Can you explain this answer?.
If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equal toa)- (3 A2 + 2 A + 5I)b)3 A2 + 2 A + 5Ic)- 3 A2 - 2 A - 5d)none of theseCorrect answer is option 'C'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equal toa)- (3 A2 + 2 A + 5I)b)3 A2 + 2 A + 5Ic)- 3 A2 - 2 A - 5d)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equal toa)- (3 A2 + 2 A + 5I)b)3 A2 + 2 A + 5Ic)- 3 A2 - 2 A - 5d)none of theseCorrect answer is option 'C'. Can you explain this answer?.
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