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A and B are two given matrices such that...

A and B are two given matrices such that the order of A is `3 xx 4` if A'Band BA' are both defined then

A

order of B' is `3xx4`

B

order of B'A is `4xx4`

C

order of B'A is `3xx3`

D

B'A is not defined

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the given matrices A and B, their orders, and the conditions under which the products A'B and BA' are defined. ### Step 1: Identify the order of matrix A The order of matrix A is given as \(3 \times 4\). This means: - A has 3 rows and 4 columns. ### Step 2: Determine the order of matrix A' The transpose of matrix A, denoted as A', will have its rows and columns swapped. Therefore, the order of A' will be: - Order of A' = \(4 \times 3\) (4 rows and 3 columns). ### Step 3: Analyze the condition A'B is defined For the product A'B to be defined, the number of columns in A' must equal the number of rows in B. - Since A' has 3 columns, B must have 3 rows. - Let the order of B be \(3 \times n\) (3 rows and n columns). ### Step 4: Analyze the condition BA' is defined For the product BA' to be defined, the number of columns in B must equal the number of rows in A'. - B has n columns, and A' has 4 rows. - Therefore, we need \(n = 4\). ### Step 5: Determine the order of matrix B From the previous steps, we have established that: - Order of B = \(3 \times 4\) (3 rows and 4 columns). ### Step 6: Determine the order of matrix B' The transpose of matrix B, denoted as B', will have its rows and columns swapped. Therefore, the order of B' will be: - Order of B' = \(4 \times 3\) (4 rows and 3 columns). ### Step 7: Check the options 1. **Option 1**: Order of B' is \(3 \times 4\) - **Incorrect** (we found it to be \(4 \times 3\)). 2. **Option 2**: Order of B'A is \(4 \times 4\) - **Correct** (B' is \(4 \times 3\) and A is \(3 \times 4\), so \(4 \times 4\)). 3. **Option 3**: Order of B'A is \(3 \times 3\) - **Incorrect**. 4. **Option 4**: B'A is not defined - **Incorrect** (it is defined). ### Conclusion The correct answer is **Option 2**: The order of B'A is \(4 \times 4\).
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