Question

If A, B are square matrices of order 3, A is non – singular and AB = 0, then B is a
(A). Null matrix
(B). Singular matrix
(C). Unit matrix
(D). Non – singular matrix

Answer 1

Hint: As we are given that AB = 0 and also A is non – singular so we can take ${{A}^{-1}}$ to both the side to make $B=0{{A}^{-1}}$ and also any matrix multiplied by null matrix will become a null matrix.

Complete step-by-step solution -
In the given question A, B are square matrices given which are said that A is non – singular and also AB is equal to 0, we have to tell about matrix B.
Before proceeding let’s gain some knowledge about matrices.
A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly a matrix over a field F is a rectangular array of scalar each of which is a member of F. The members, symbols, or expressions are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns respectively.
If the number of rows and columns of a matrix is equal then it is said as a square matrix.
A non – singular matrix is a square one whose determinant is non – zero.
The ‘0’ means the null matrix or whose determinant is 0.
So, we are given AB = 0 which we can also write as $B=0{{A}^{-1}}$ where ${{A}^{-1}}$ is inverse of matrix A and we know that null matrix multiplied by any matrix will be considered as null.
So, B will be considered as a null matrix.
Hence, the correct option is (a).

Note: We can also say that B is a null matrix by telling that its product is the null matrix of A and B then either of two matrices should be a null matrix as A is non – singular hence B should be the null matrix.