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If A is a singular matrix, then A adj (A)
- a)is an identity matrix
- b)is a zero scalar
- c)is a scalar matrix
- d)is an orthogonal matrix
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If A is a singular matrix, then A adj (A)a)is an identity matrixb)is a...
Explanation:
To understand why A adj(A) is a zero scalar matrix when A is a singular matrix, let us first understand what is a singular matrix and what is an adjoint matrix.
Singular Matrix:
A matrix is said to be singular if it does not have an inverse. That is, if A is a square matrix of order n, then A is said to be singular if det(A) = 0.
Adjoint Matrix:
The adjoint matrix of a square matrix A of order n is denoted by adj(A) and is defined as the transpose of the matrix of co-factors of A. That is,
adj(A) = [C]T
where C is the matrix of co-factors of A.
Now, let us consider the product A adj(A), where A is a singular matrix.
A adj(A) = A [C]T
= [AC]T
= [det(A) I]T
= det(A) I
where I is the identity matrix of order n.
Since A is a singular matrix, det(A) = 0.
Therefore, A adj(A) = 0 I = 0.
Hence, the correct option is B: A adj(A) is a zero scalar matrix when A is a singular matrix.
To understand why A adj(A) is a zero scalar matrix when A is a singular matrix, let us first understand what is a singular matrix and what is an adjoint matrix.
Singular Matrix:
A matrix is said to be singular if it does not have an inverse. That is, if A is a square matrix of order n, then A is said to be singular if det(A) = 0.
Adjoint Matrix:
The adjoint matrix of a square matrix A of order n is denoted by adj(A) and is defined as the transpose of the matrix of co-factors of A. That is,
adj(A) = [C]T
where C is the matrix of co-factors of A.
Now, let us consider the product A adj(A), where A is a singular matrix.
A adj(A) = A [C]T
= [AC]T
= [det(A) I]T
= det(A) I
where I is the identity matrix of order n.
Since A is a singular matrix, det(A) = 0.
Therefore, A adj(A) = 0 I = 0.
Hence, the correct option is B: A adj(A) is a zero scalar matrix when A is a singular matrix.
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If A is a singular matrix, then A adj (A)a)is an identity matrixb)is a zero scalarc)is a scalar matrixd)is an orthogonal matrixCorrect answer is option 'B'. Can you explain this answer?
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