Question

If A is a matrix of order $3\times4$ and B is a matrix such that AB and BA are both defined, then order of B is:

• A. $3\times4$
• B. $3\times3$
• C. $4\times3$
• D. $4\times4$

Hint:

For both AB and BA to exist, B must satisfy both compatibility conditions simultaneously.

Answer 1

The Correct Option is C

Concept: For matrix multiplication: - If $AB$ exists, columns of $A$ = rows of $B$ - If $BA$ exists, columns of $B$ = rows of $A$

Step 1: {Given A is $3\times4$.}

Step 2: {For $AB$ to exist, B must be $4\times n$.}

Step 3: {For $BA$ to exist, B must be $m\times3$.}

Step 4: {Combine both conditions.}
So B must be: \[ B = 4\times3 \]

Step 5: {Hence order of B is $4\times3$.}