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If A("mxn") is a matrix and I and I' are...

If `A_("mxn")` is a matrix and I and I' are unit matrices such that I'AI is defined, then which of the following is true ?

A

`t_(r)(I+l')=m+n`

B

`t_(r)(Il')=m`

C

`t_(r)(l)+t_(r)(l')=m+n`

D

None of these.

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the given information about the matrices and determine which statement is true based on the properties of matrices. ### Step-by-Step Solution: 1. **Understanding the Given Matrices**: - Let \( A \) be a matrix of size \( m \times n \). - Let \( I \) be a unit (identity) matrix of size \( m \times m \). - Let \( I' \) be a unit (identity) matrix of size \( n \times n \). 2. **Analyzing the Expression \( I'AI \)**: - The expression \( I'AI \) is defined, which means the multiplication of these matrices is valid. - For the multiplication \( I'AI \) to be defined, the number of columns in \( I' \) must equal the number of rows in \( A \) and the number of columns in \( A \) must equal the number of rows in \( I \). 3. **Determining the Sizes**: - Since \( I' \) is \( n \times n \) and \( A \) is \( m \times n \), the multiplication \( I'A \) is defined if \( n = m \) (the number of columns of \( I' \) equals the number of rows of \( A \)). - Next, \( AI \) is defined if \( n = m \) (the number of columns of \( A \) equals the number of rows of \( I \)). - Thus, we conclude that \( I'AI \) is defined when \( m \) and \( n \) are compatible in terms of matrix multiplication. 4. **Evaluating the Options**: - **Option 1**: Trace of \( I + I' \) - This is incorrect because \( I \) is \( m \times m \) and \( I' \) is \( n \times n \), and they cannot be added unless \( m = n \). - **Option 2**: \( I' + I \) - This is also incorrect for the same reason as above. - **Option 3**: Trace of \( I + I' \) - The trace of \( I \) is \( m \) and the trace of \( I' \) is \( n \). Therefore, the trace of \( I + I' \) is \( m + n \), which is valid if \( m = n \). - **Option 4**: Any other option that is not defined - This is incorrect. 5. **Conclusion**: - The correct answer is that the trace of \( I + I' \) is \( m + n \). ### Final Answer: The correct option is that the trace of \( I + I' \) is \( m + n \).
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