Question Bank - Mathematics

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Consider the following statements in respect of square matrices A and B of same order :1. If AB is a null matrix, then at least one of A and B is a null matrix.2. If AB is an identity matrix, then BA = AB.Which of the above statements is/are correct?

A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2

Solution:

Concept:1). The null matrix is a matrix having zero as all its elements. The null matrix is also called a zero matrix.2). An identity matrix is a given square matrix of any order which contains on its main diagonal elements with value of oneCalculation:Statement I: If AB is a null matrix, then at least one of A and B is a null matrix.Let us consider two matrices,\(A = \begin{bmatrix} 0 &0 \\[0.3em] 0 & 1\\[0.3em] \end{bmatrix}\ and\ B = \begin{bmatrix} 0 &1 \\[0.3em] 0 &0\\[0.3em] \end{bmatrix} \)AB = \( \begin{bmatrix} 0 &0 \\[0.3em] 0 & 1\\[0.3em] \end{bmatrix}\ \begin{bmatrix} 0 &1 \\[0.3em] 0 &0\\[0.3em] \end{bmatrix}= \begin{bmatrix} 0 &0 \\[0.3em] 0 & 0\\[0.3em] \end{bmatrix}\ \)Here, both A and B are non-zero matrices. But the product is zero matrices.? Statement I is incorrect.Statement II: If AB is an identity matrix, then BA = AB.AB = IPre-multiplying A-1 on both sides.? A-1AB = A-1I [Considering A is invertible]? B = A-1Post-multiplying A on both sides.? BA = A-1A? BA = I? Statement II is correct.? Only statement II is correct.

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