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Suppose Q is a 3 by 3 metrix of rank 2 and T:M(R) - M(R) be the linear transformation defined by T(P) =QP. Then the rank of T is? And M(R) is 3*3 metrix?
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Suppose Q is a 3 by 3 metrix of rank 2 and T:M(R) - M(R) be the linear...
**Rank of Q**
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. Since Q is a 3 by 3 matrix of rank 2, it means that there are only two linearly independent rows or columns in Q.
**Linear Transformation T**
The linear transformation T is defined as T(P) = QP, where P is a matrix in M(R) - the set of all 3 by 3 matrices over the field of real numbers.
**Rank of T**
To determine the rank of T, we need to find the maximum number of linearly independent rows or columns in the matrix representation of T.
Let's consider the matrix representation of T. Since P is a 3 by 3 matrix, and Q is a 3 by 3 matrix of rank 2, the product QP will also be a 3 by 3 matrix. Let's represent QP as R.
R = QP
Now, let's analyze the structure of R. Since Q is a matrix of rank 2, it means that there are only two linearly independent rows or columns in Q. Therefore, when we multiply Q by P, the resulting matrix R will have at most two linearly independent rows or columns.
However, it is possible that the product QP could have fewer linearly independent rows or columns than Q itself. This is because the multiplication process may introduce linear dependencies among the rows or columns of Q.
Therefore, the rank of T, which is the maximum number of linearly independent rows or columns in the matrix representation of T, is at most 2. It cannot be greater than the rank of Q.
In conclusion, the rank of T is at most 2.
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. Since Q is a 3 by 3 matrix of rank 2, it means that there are only two linearly independent rows or columns in Q.
**Linear Transformation T**
The linear transformation T is defined as T(P) = QP, where P is a matrix in M(R) - the set of all 3 by 3 matrices over the field of real numbers.
**Rank of T**
To determine the rank of T, we need to find the maximum number of linearly independent rows or columns in the matrix representation of T.
Let's consider the matrix representation of T. Since P is a 3 by 3 matrix, and Q is a 3 by 3 matrix of rank 2, the product QP will also be a 3 by 3 matrix. Let's represent QP as R.
R = QP
Now, let's analyze the structure of R. Since Q is a matrix of rank 2, it means that there are only two linearly independent rows or columns in Q. Therefore, when we multiply Q by P, the resulting matrix R will have at most two linearly independent rows or columns.
However, it is possible that the product QP could have fewer linearly independent rows or columns than Q itself. This is because the multiplication process may introduce linear dependencies among the rows or columns of Q.
Therefore, the rank of T, which is the maximum number of linearly independent rows or columns in the matrix representation of T, is at most 2. It cannot be greater than the rank of Q.
In conclusion, the rank of T is at most 2.
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Suppose Q is a 3 by 3 metrix of rank 2 and T:M(R) - M(R) be the linear transformation defined by T(P) =QP. Then the rank of T is? And M(R) is 3*3 metrix?
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Suppose Q is a 3 by 3 metrix of rank 2 and T:M(R) - M(R) be the linear transformation defined by T(P) =QP. Then the rank of T is? And M(R) is 3*3 metrix? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Suppose Q is a 3 by 3 metrix of rank 2 and T:M(R) - M(R) be the linear transformation defined by T(P) =QP. Then the rank of T is? And M(R) is 3*3 metrix? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose Q is a 3 by 3 metrix of rank 2 and T:M(R) - M(R) be the linear transformation defined by T(P) =QP. Then the rank of T is? And M(R) is 3*3 metrix?.
Suppose Q is a 3 by 3 metrix of rank 2 and T:M(R) - M(R) be the linear transformation defined by T(P) =QP. Then the rank of T is? And M(R) is 3*3 metrix? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Suppose Q is a 3 by 3 metrix of rank 2 and T:M(R) - M(R) be the linear transformation defined by T(P) =QP. Then the rank of T is? And M(R) is 3*3 metrix? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose Q is a 3 by 3 metrix of rank 2 and T:M(R) - M(R) be the linear transformation defined by T(P) =QP. Then the rank of T is? And M(R) is 3*3 metrix?.
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