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If A, B are, respectively m × n, k × l matrices, then both AB and BA are defined if and only if
- a)n = m , k = m
- b)m = l , n= l
- c)m = n, k = l
- d)n = k and l = m.
Correct answer is option 'D'. Can you explain this answer?
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If A, B are, respectively m × n, k × l matrices, then both A...
If A, B are, respectively m × n, k × l matrices, then both AB and BA are defined if and only if n = k and l = m. In particular, if both A and B are square matrices of the same order, then both AB and BA are defined.
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If A, B are, respectively m × n, k × l matrices, then both A...
Explanation:
The question is about the conditions required for the matrices A and B to be multiplied in both orders, i.e., AB and BA.
To understand this, we need to recall the rules of matrix multiplication.
In general, for two matrices A and B to be multiplied, the number of columns in A must equal the number of rows in B.
So, if A is an m x n matrix, and B is a k x l matrix, then we need k = n for AB to be defined and l = m for BA to be defined.
Let's consider the given options one by one:
A) n = m , k = m
If n = m, then AB is defined, but for BA to be defined, we need l = n, which is not given. So, this option is not correct.
B) m = l , n= k
If m = l, then BA is defined, but for AB to be defined, we need k = n, which is not given. So, this option is not correct.
C) m = n, k = l
If m = n and k = l, then both AB and BA are defined. So, this option is correct.
D) n = k and l = m
If n = k, then AB is defined, but for BA to be defined, we need l = n, which is not given. So, this option is not correct.
Therefore, the correct answer is option C, i.e., m = n, k = l.
The question is about the conditions required for the matrices A and B to be multiplied in both orders, i.e., AB and BA.
To understand this, we need to recall the rules of matrix multiplication.
In general, for two matrices A and B to be multiplied, the number of columns in A must equal the number of rows in B.
So, if A is an m x n matrix, and B is a k x l matrix, then we need k = n for AB to be defined and l = m for BA to be defined.
Let's consider the given options one by one:
A) n = m , k = m
If n = m, then AB is defined, but for BA to be defined, we need l = n, which is not given. So, this option is not correct.
B) m = l , n= k
If m = l, then BA is defined, but for AB to be defined, we need k = n, which is not given. So, this option is not correct.
C) m = n, k = l
If m = n and k = l, then both AB and BA are defined. So, this option is correct.
D) n = k and l = m
If n = k, then AB is defined, but for BA to be defined, we need l = n, which is not given. So, this option is not correct.
Therefore, the correct answer is option C, i.e., m = n, k = l.
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