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If A and B are two matrices such that A + B and AB are both defined, then A and B are
- a)both null matrices
- b)both identity matrices
- c)both square matrices of the same order
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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**Explanation:**
To understand why the correct answer is option 'C', let's first define what the terms mean:
1. **Null matrix:** A null matrix is a matrix in which every element is zero.
2. **Identity matrix:** An identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros. The principal diagonal is the set of elements where the row index and column index are the same.
3. **Square matrix:** A square matrix is a matrix in which the number of rows is equal to the number of columns.
Now, let's analyze the given statement: "If A and B are two matrices such that AB and BA are both defined."
If AB is defined, it means that the number of columns in matrix A is equal to the number of rows in matrix B. Let's say A is an m x n matrix and B is an n x p matrix. The product of these matrices will be an m x p matrix.
Similarly, if BA is defined, it means that the number of columns in matrix B is equal to the number of rows in matrix A. Let's say B is a p x q matrix and A is a q x m matrix. The product of these matrices will be a p x m matrix.
So, from the given statement, we can conclude that:
- The number of columns in A is equal to the number of rows in B.
- The number of columns in B is equal to the number of rows in A.
From these conclusions, we can deduce that:
- The number of columns in A is equal to the number of rows in B, i.e., n = p.
- The number of columns in B is equal to the number of rows in A, i.e., p = q.
Therefore, the number of columns in A is equal to the number of rows in B, and the number of columns in B is equal to the number of rows in A. This implies that both A and B are square matrices of the same order.
Hence, the correct answer is option 'C' - Both matrices A and B are square matrices of the same order.
To understand why the correct answer is option 'C', let's first define what the terms mean:
1. **Null matrix:** A null matrix is a matrix in which every element is zero.
2. **Identity matrix:** An identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros. The principal diagonal is the set of elements where the row index and column index are the same.
3. **Square matrix:** A square matrix is a matrix in which the number of rows is equal to the number of columns.
Now, let's analyze the given statement: "If A and B are two matrices such that AB and BA are both defined."
If AB is defined, it means that the number of columns in matrix A is equal to the number of rows in matrix B. Let's say A is an m x n matrix and B is an n x p matrix. The product of these matrices will be an m x p matrix.
Similarly, if BA is defined, it means that the number of columns in matrix B is equal to the number of rows in matrix A. Let's say B is a p x q matrix and A is a q x m matrix. The product of these matrices will be a p x m matrix.
So, from the given statement, we can conclude that:
- The number of columns in A is equal to the number of rows in B.
- The number of columns in B is equal to the number of rows in A.
From these conclusions, we can deduce that:
- The number of columns in A is equal to the number of rows in B, i.e., n = p.
- The number of columns in B is equal to the number of rows in A, i.e., p = q.
Therefore, the number of columns in A is equal to the number of rows in B, and the number of columns in B is equal to the number of rows in A. This implies that both A and B are square matrices of the same order.
Hence, the correct answer is option 'C' - Both matrices A and B are square matrices of the same order.
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If A and B are two matrices such that A + B and AB are both defined, then A and B area)both null matricesb)both identity matricesc)both square matrices of the same orderd)None of theseCorrect answer is option 'C'. Can you explain this answer?
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