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The rank of a 3×3 matrix C (=AB), found by multiplying a non-zero column matrix Aof size 3×1 and a non-zero row matrix B of size 1×3, is
- a)0
- b)1
- c)2
- d)3
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
The rank of a 3×3 matrix C (=AB), found by multiplying a non-zero...
(b)
Let A = 
Then C = AB =
Then det (AB) = 0.
Then also every minor
of order 2 is also zero.
∴ rank(C) =1.
of order 2 is also zero.
∴ rank(C) =1.
Most Upvoted Answer
The rank of a 3×3 matrix C (=AB), found by multiplying a non-zero...
**Rank of a Matrix**
The rank of a matrix refers to the maximum number of linearly independent rows or columns in the matrix. It is denoted by 'r'.
**Matrix Multiplication**
Matrix multiplication is a binary operation that produces a matrix from two matrices. In order for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
**Given Information**
In this question, we are given that matrix C is obtained by multiplying a non-zero column matrix A of size 31 and a non-zero row matrix B of size 13. The size of matrix C is not explicitly mentioned.
**Size of Matrix C**
Since matrix A is a column matrix of size 31 and matrix B is a row matrix of size 13, the resulting matrix C will have dimensions (31 x 1) * (1 x 13) = (31 x 13).
**Rank of Matrix C**
To find the rank of matrix C, we need to determine the maximum number of linearly independent rows or columns in matrix C.
Since matrix C is a 31 x 13 matrix, the maximum possible rank is min(31, 13) = 13. This means that the maximum rank of matrix C can be 13.
**Conclusion**
According to the given information, the rank of matrix C (obtained by multiplying matrix A and matrix B) is 1.
Therefore, the correct answer is option 'B' - 1.
The rank of a matrix refers to the maximum number of linearly independent rows or columns in the matrix. It is denoted by 'r'.
**Matrix Multiplication**
Matrix multiplication is a binary operation that produces a matrix from two matrices. In order for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
**Given Information**
In this question, we are given that matrix C is obtained by multiplying a non-zero column matrix A of size 31 and a non-zero row matrix B of size 13. The size of matrix C is not explicitly mentioned.
**Size of Matrix C**
Since matrix A is a column matrix of size 31 and matrix B is a row matrix of size 13, the resulting matrix C will have dimensions (31 x 1) * (1 x 13) = (31 x 13).
**Rank of Matrix C**
To find the rank of matrix C, we need to determine the maximum number of linearly independent rows or columns in matrix C.
Since matrix C is a 31 x 13 matrix, the maximum possible rank is min(31, 13) = 13. This means that the maximum rank of matrix C can be 13.
**Conclusion**
According to the given information, the rank of matrix C (obtained by multiplying matrix A and matrix B) is 1.
Therefore, the correct answer is option 'B' - 1.
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The rank of a 3×3 matrix C (=AB), found by multiplying a non-zero column matrix Aof size 3×1 and a non-zero row matrix B of size 1×3, isa)0b)1c)2d)3Correct answer is option 'B'. Can you explain this answer?
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