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Let A be a 3 x 3 real matrix such that A2 = -I3 where I3 is the 3 x 3 identity matrix then a matrix A is
- a)diagonalizable
- b)Orthogonal
- c)does not exist
- d)symmetric
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Let A be a 3 x 3 real matrix such that A2 = -I3 where I3 is the 3 x 3 ...
Explanation:
To understand why the correct answer is option 'C' (the matrix does not exist), let's consider the given information and analyze it step by step.
1. We are given that A is a 3 x 3 real matrix.
2. We are also given that A^2 = -I3, where I3 is the 3 x 3 identity matrix.
Analysis:
To find the matrix A, we need to square it and equate it to -I3. Let's calculate A^2.
A = [a11 a12 a13]
[a21 a22 a23]
[a31 a32 a33]
A^2 = A * A
= [a11 a12 a13] * [a11 a12 a13]
[a21 a22 a23] [a21 a22 a23]
[a31 a32 a33] [a31 a32 a33]
Multiplying the two matrices, we get:
A^2 = [a11*a11 + a12*a21 + a13*a31 a11*a12 + a12*a22 + a13*a32 a11*a13 + a12*a23 + a13*a33]
[a21*a11 + a22*a21 + a23*a31 a21*a12 + a22*a22 + a23*a32 a21*a13 + a22*a23 + a23*a33]
[a31*a11 + a32*a21 + a33*a31 a31*a12 + a32*a22 + a33*a32 a31*a13 + a32*a23 + a33*a33]
Comparing this with -I3, we have:
a11*a11 + a12*a21 + a13*a31 = -1 (1)
a11*a12 + a12*a22 + a13*a32 = 0 (2)
a11*a13 + a12*a23 + a13*a33 = 0 (3)
a21*a11 + a22*a21 + a23*a31 = 0 (4)
a21*a12 + a22*a22 + a23*a32 = -1 (5)
a21*a13 + a22*a23 + a23*a33 = 0 (6)
a31*a11 + a32*a21 + a33*a31 = 0 (7)
a31*a12 + a32*a22 + a33*a32 = 0 (8)
a31*a13 + a32*a23 + a33*a33 = -1 (9)
Analysis of the System of Equations:
From equations (2) and (6), we can conclude that a11*a12 = 0 and a21*a13 = 0. This means that either a11 = 0 or a12 = 0, and either a21 = 0 or a13 = 0.
Similarly, from equations (4) and (8), we can conclude that a21*a12 = 0 and a31*a12 = 0. This means that either a21 = 0 or a12 = 0, and either a31 = 0 or a12 = 0.
However, these conclusions lead to contradictions since a
To understand why the correct answer is option 'C' (the matrix does not exist), let's consider the given information and analyze it step by step.
1. We are given that A is a 3 x 3 real matrix.
2. We are also given that A^2 = -I3, where I3 is the 3 x 3 identity matrix.
Analysis:
To find the matrix A, we need to square it and equate it to -I3. Let's calculate A^2.
A = [a11 a12 a13]
[a21 a22 a23]
[a31 a32 a33]
A^2 = A * A
= [a11 a12 a13] * [a11 a12 a13]
[a21 a22 a23] [a21 a22 a23]
[a31 a32 a33] [a31 a32 a33]
Multiplying the two matrices, we get:
A^2 = [a11*a11 + a12*a21 + a13*a31 a11*a12 + a12*a22 + a13*a32 a11*a13 + a12*a23 + a13*a33]
[a21*a11 + a22*a21 + a23*a31 a21*a12 + a22*a22 + a23*a32 a21*a13 + a22*a23 + a23*a33]
[a31*a11 + a32*a21 + a33*a31 a31*a12 + a32*a22 + a33*a32 a31*a13 + a32*a23 + a33*a33]
Comparing this with -I3, we have:
a11*a11 + a12*a21 + a13*a31 = -1 (1)
a11*a12 + a12*a22 + a13*a32 = 0 (2)
a11*a13 + a12*a23 + a13*a33 = 0 (3)
a21*a11 + a22*a21 + a23*a31 = 0 (4)
a21*a12 + a22*a22 + a23*a32 = -1 (5)
a21*a13 + a22*a23 + a23*a33 = 0 (6)
a31*a11 + a32*a21 + a33*a31 = 0 (7)
a31*a12 + a32*a22 + a33*a32 = 0 (8)
a31*a13 + a32*a23 + a33*a33 = -1 (9)
Analysis of the System of Equations:
From equations (2) and (6), we can conclude that a11*a12 = 0 and a21*a13 = 0. This means that either a11 = 0 or a12 = 0, and either a21 = 0 or a13 = 0.
Similarly, from equations (4) and (8), we can conclude that a21*a12 = 0 and a31*a12 = 0. This means that either a21 = 0 or a12 = 0, and either a31 = 0 or a12 = 0.
However, these conclusions lead to contradictions since a
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Let A be a 3 x 3 real matrix such that A2 = -I3 where I3 is the 3 x 3 identity matrix then a matrix A isa)diagonalizableb)Orthogonalc)does not existd)symmetricCorrect answer is option 'C'. Can you explain this answer?
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