Question

If A is a singular matrix, then adj A is

A. non-singular

B. singular

C. symmetric

D. not defined

Answer 1

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:

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.