If A is a singular matrix, then adj A isA. non-singularB. singularC. symmetricD. not defined
Answer
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Hint: Here, given that matrix A is singular hence its determinant is equal to 0. Also apply the formula which relates determinant of A and adjoint of A to get the result.
Complete step-by-step answer:
So, the correct answer is “Option B”.
Note: In these types of questions, apply the basic concept of singular matrix and related formula. Also keep in mind if a product of two numbers is 0 and one the number is non-zero then the second number must be 0.
Complete step-by-step answer:
Singular Matrix: A singular matrix means a matrix which is non-invertible i.e. there is no multiplicative inverse or no inverse exists for that matrix.
Therefore, a matrix is singular if and only if its determinant is zero.
⇒Determinant of A is equal to 0 i.e., ∣A∣ = 0
Determinant: Determinant of a matrix is a value associated with that matrix which can be found by using a particular method. Also determinant is possible for square matrices only.
Also we know that, A(adjA) = ∣A∣ × I = 0 × I = 0,
where I is an identity matrix having the same order as A and adj A represents an adjoint matrix of A.
∴ A(adjA) is a zero matrix.
So, the correct answer is “Option B”.
Note: In these types of questions, apply the basic concept of singular matrix and related formula. Also keep in mind if a product of two numbers is 0 and one the number is non-zero then the second number must be 0.
Alternatively this can be remembered as a statement if a matrix is singular then its adjoint will be 0 and if the adjoint of any matrix is zero then that matrix will be zero. This statement can be directly used as a standard definition or rule.
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