Question

If the product of two non-null square matrices A and B is a null-matrix. What will be the characteristics of given matrices A and B.
(a) Only one of them is singular
(b) both of them must be singular.
(c) both of them are non-singular
(d) none of these.

Answer 1

Hint: We have two matrices A and B whose product A.B=0. We use properties of singular, non-singular and null matrices to solve this problem. We assume one of the matrices as non-singular and find the characteristics for another matrix. We come to a conclusion about both matrices once we proceed through this.

Complete step-by-step answer:
Given that we have two non-null square matrices A and B and the product of A and B is a null-matrix.
Null matrix: If every element in the matrix is 0, then it is known as null matrix.
Square matrix: If the no. of rows is equal to the no. of columns, then it is known as square matrix.
According to the problem, $ A\ne 0 $ and $ B\ne 0 $ but, $ A.B=0 $ . We need to find the characteristics of both matrices A and B.
Now we find characteristics by assuming one of the given matrices as non-singular matrix and check what happens with the other matrix.
Singular matrix: If the determinant of a given matrix is equal to 0, then the given matrix is known as a singular matrix. Inverse doesn’t exist for singular matrices.
Non-singular matrix: If the determinant of a given matrix is not equal to 0, then the given matrix is known as a non-singular matrix. Inverse exists for non-singular matrices.
If $ X $ is a nonsingular matrix and its inverse is $ {{X}^{-1}} $ , then $ X.{{X}^{-1}}={{X}^{-1}}.X=I $ . Where ‘I’ is an identity matrix.
Let us assume B is a nonsingular matrix and its inverse is $ {{B}^{-1}} $ .
We have $ A.B=0 $ .
Now we multiply $ {{B}^{-1}} $ on both sides,
 $ A.B.{{B}^{-1}}=0.{{B}^{-1}} $ .
We know that $ B.{{B}^{-1}}={{B}^{-1}}.B=I $ and any matrix multiplied with a null matrix gives a null matrix.
 $ A.I=0 $ .
We know that any matrix multiplied with an Identity matrix gives the same matrix.
 $ A=0 $ .
Here we got ‘A’ as a null matrix. But, according to the problem ‘A’ is not a null-matrix. So, the inverse doesn’t exist for ‘B’. This makes ‘B’ a singular matrix.
Let us assume A is a nonsingular matrix and its inverse is $ {{A}^{-1}} $ .
We have $ A.B=0 $ .
Now we multiply $ {{A}^{-1}} $ on both sides,
 $ {{A}^{-1}}.A.B={{A}^{-1}}.0 $ .
We know that $ {{A}^{-1}}.A=A.{{A}^{-1}}=I $ and any matrix multiplied with a null matrix gives a null matrix.
 $ I.B=0 $ .
We know that any matrix multiplied with an Identity matrix gives the same matrix.
 $ B=0 $ .
Here we got ‘B’ as a null matrix. But, according to the problem ‘B’ is not a null-matrix. So, the inverse doesn’t exist for ‘A’. This makes ‘A’ a singular matrix.
We came to a conclusion that both matrices ‘A’ and ‘B’ must be singular.
So, the correct answer is “Option B”.

Note: Alternatively we can apply determinant for the product $ AB=0 $ and use the property $ \left| AB \right|=\left| A \right|.\left| B \right| $ to prove either $ \left| A \right|=0 or \left| B \right|=0 $ or both. This alternative method may not give an absolute answer if the options are given as at least one of the matrices are singular. We follow by assuming one of the matrices is nonsingular whenever we find such types of problems.