If A and B are two matrices such that AB is of order n × n, then which one of the following is correct?
Orders of A and B need not be the same.
Understanding Matrix Multiplication and Orders
The question asks about the relationship between the orders (dimensions) of two matrices, A and B, given that their product AB is defined and has a specific order, n × n.
For matrix multiplication AB to be defined, the number of columns in the first matrix (A) must be equal to the number of rows in the second matrix (B).
Let the order of matrix A be \(m \times p\). This means A has \(m\) rows and \(p\) columns.
Let the order of matrix B be \(q \times r\). This means B has \(q\) rows and \(r\) columns.
For the product AB to be defined, the number of columns in A must equal the number of rows in B. So, we must have \(p = q\).
If AB is defined, the resulting matrix AB will have the number of rows from the first matrix (A) and the number of columns from the second matrix (B). Thus, the order of AB is \(m \times r\).
The question states that the product AB is of order \(n \times n\).
Comparing the order of AB (\(m \times r\)) with the given order (\(n \times n\)), we find that:
- The number of rows in AB is \(m\), which must equal \(n\). So, \(m=n\).
- The number of columns in AB is \(r\), which must equal \(n\). So, \(r=n\).
Putting it all together, for AB to be an \(n \times n\) matrix, the orders of A and B must satisfy the following conditions:
- Order of A is \(m \times p\), where \(m=n\). So, A is \(n \times p\).
- Order of B is \(q \times r\), where \(q=p\) (for multiplication to be defined) and \(r=n\). So, B is \(p \times n\).
Thus, if A is an \(n \times p\) matrix and B is a \(p \times n\) matrix for any positive integer \(p\), their product AB will be an \(n \times n\) matrix.
Now let's examine the given options:
- Option 1: A and B should be square matrices of same order. This implies A is \(k \times k\) and B is \(k \times k\) for some \(k\). For AB to be \(n \times n\), \(k\) must be \(n\). So, A is \(n \times n\) and B is \(n \times n\). This is a specific case where \(p=n\) in our derived orders (\(n \times p\) and \(p \times n\)). It is not a necessary condition; for example, if A is \(2 \times 3\) and B is \(3 \times 2\), AB is \(2 \times 2\). Here \(n=2\) and \(p=3\), and neither A nor B is square of the same order.
- Option 2: Either A or B should be a square matrix. As shown in the example above (A is \(2 \times 3\), B is \(3 \times 2\), AB is \(2 \times 2\)), neither A nor B needs to be a square matrix for AB to be an \(n \times n\) matrix (here \(n=2\)). This option is incorrect.
- Option 3: Both A and B should be of same order. Let the order be \(k \times l\). For AB to be defined, \(l=k\). So both must be \(k \times k\). Their product AB would be \(k \times k\). If AB is \(n \times n\), then \(k=n\). This implies A is \(n \times n\) and B is \(n \times n\). Again, this is a specific case (\(p=n\)) and not a necessary condition.
- Option 4: Orders of A and B need not be the same. We found that A is \(n \times p\) and B is \(p \times n\). The order of A is \((n, p)\) and the order of B is \((p, n)\). These orders are the same only if \(n=p\). If \(n \neq p\), the orders are different (e.g., A is \(2 \times 3\), B is \(3 \times 2\), orders are \((2,3)\) and \((3,2)\) which are not the same). Therefore, the orders of A and B need not be the same. This statement is correct.
Let's summarize the matrix multiplication rule for orders:
| Matrix | Order (Rows × Columns) |
|---|---|
| A | \(m \times p\) |
| B | \(q \times r\) |
| Condition for AB to be defined | \(p = q\) |
| Order of AB (if defined) | \(m \times r\) |
Given that the order of AB is \(n \times n\), we have \(m=n\) and \(r=n\). With \(p=q\), the orders are:
- Order of A: \(n \times p\)
- Order of B: \(p \times n\)
For these orders to be the same, we need \(n=p\) and \(p=n\), which implies \(n=p\). However, \(p\) can be any positive integer. If \(p \neq n\), the orders are different. Therefore, the orders of A and B need not be the same.
Revision Table: Key Concepts in Matrix Multiplication
| Concept | Description | Rule for Orders |
|---|---|---|
| Matrix Definition | A rectangular array of numbers. | Order: rows \(\times\) columns (\(m \times n\)) |
| Matrix Addition/Subtraction | Possible only if matrices have the same order. | If A is \(m \times n\) and B is \(m \times n\), A \(\pm\) B is \(m \times n\). |
| Scalar Multiplication | Multiplying a matrix by a single number (scalar). | If A is \(m \times n\) and \(k\) is a scalar, \(kA\) is \(m \times n\). |
| Matrix Multiplication (AB) | Defined if columns of A = rows of B. | If A is \(m \times p\) and B is \(p \times r\), AB is \(m \times r\). |
Additional Information on Matrix Orders
The order of a matrix is crucial for determining which operations can be performed. For matrix multiplication AB, the 'inner' dimensions must match (columns of A and rows of B), and the 'outer' dimensions determine the order of the resulting product (rows of A and columns of B).
In the case where AB is an \(n \times n\) matrix:
- Matrix A has dimensions \(n \times p\) for some \(p \ge 1\).
- Matrix B has dimensions \(p \times n\) for the same \(p\).
If \(p=n\), both matrices A and B are \(n \times n\) square matrices, and their orders are the same. However, if \(p\) is any other positive integer not equal to \(n\), the orders \((n \times p)\) and \((p \times n)\) are different. For example, if \(n=2\) and \(p=3\), A is \(2 \times 3\) and B is \(3 \times 2\). AB is \(2 \times 2\). The orders of A and B are clearly not the same.
This demonstrates that for the product AB to be an \(n \times n\) matrix, the orders of A and B are not necessarily the same. They only need to follow the pattern \(n \times p\) and \(p \times n\).
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Select the correct answer using the code given below:
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