Question

Q.No.3 If A and B are matrices such
that A+B and BA are both defined,
then
1.A and B can be any matrices

2.A, B are square matrices not
necessarily of same order

3.A, B are square matrices of same order

4.number of columns of A= number of
rows of B​

Answer 1

$\underline{\textsf{Given:}}$

A+B and AB are defined

$\underline{\textsf{To find:}}$

$\textsf{Most appropriate answer in the given alternatives}$

$\underline{\textsf{Solution:}}$

Since A+B is defined, we have

$\textsf{A and B must be of same order}$.......(1)

Since AB is defined, we have

$\textsf{Number of rows of A must be number f columns of B}$.......(2)

$\textsf{A and B must satisfy conditions (1) and (2)}$

$\textsf{The only possibility is}$

$\textsf{A and B must be square matrices of same order}$

$\underline{\textsf{Answer:}}$

$\textsf{Option (3) is correct}$

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

1. The sum of two matrices A & B is defined when when A & B are of same order

2. Two matrices can be multiplied only when

Number of columns of 1st = Number of rows of 2nd

GIVEN

A and B are matrices such

A and B are matrices suchthat A+B and BA are both defined

TO CHOOSE THE RIGHT OPTION

1. A and B can be any matrices

2. A, B are square matrices not necessarily of same order

3. A, B are square matrices of same order

4. Number of columns of A= number of

CALCULATION

Let order of A is m × n and order of B is p × q

Since A + B is defined

So A & B are of same order

Hence

$\sf{ \: p = m \: \: \: \: \: and \: \: \: \: q = n\: }$

So order of both A & B are m × n

Again BA is defined

So

Number of columns of B = Number of rows of A

Hence n = m

So order of both A & B are m × m

So A & B both are square matrices of same order

RESULT

The right option is

3. A, B are square matrices of same order