Matrix Multiplication and Rank of Matrix

In summary, Matrix multiplication is an operation performed on two matrices to produce a new matrix by multiplying each element in a row of the first matrix by the corresponding element in a column of the second matrix and then summing the products. The rank of a matrix is determined by the number of linearly independent rows or columns and is important for providing information about the system of equations represented by the matrix. The rank of a matrix can change after performing matrix multiplication and is not commutative, meaning the order of multiplication matters.
  • #1
rajtendulkar
2
0
Dear Forum,

I have one question on matrix multiplication.

Suppose there are 2 matrices -

A = 1 -1 0
0 2 -1
2 0 -1

B = 1
1
2

and AB = 0 (Zero Matrix)
if B not a zero-matrix, then rank(A) is less than s, where s is the dimension of B.

I wanted to have a proof and explanation for this.
Any book / link? :)

Thank You,
Raj
 
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  • #3
Thanks a lot for the reply ! :)
 

1. What is matrix multiplication and how is it performed?

Matrix multiplication is an operation performed on two matrices to produce a new matrix. It involves multiplying each element in a row of the first matrix by the corresponding element in a column of the second matrix, and then summing the products. This process is repeated for each row and column combination to fill in the new matrix.

2. How is the rank of a matrix determined?

The rank of a matrix is determined by the number of linearly independent rows or columns in the matrix. In other words, it is the maximum number of rows or columns that can be combined to form a linearly independent set. This can be found using various methods such as Gaussian elimination or calculating the determinant.

3. What is the significance of the rank of a matrix?

The rank of a matrix is important because it provides information about the system of equations represented by the matrix. A matrix with full rank (i.e. rank equal to the number of rows or columns) indicates that the equations are consistent and have a unique solution. A matrix with rank less than the number of rows or columns may have infinitely many solutions or no solutions at all.

4. Can the rank of a matrix change after performing matrix multiplication?

Yes, the rank of a matrix can change after performing matrix multiplication. In fact, the rank of the resulting matrix will always be equal to or less than the minimum rank of the two original matrices. For example, if one matrix has rank 3 and the other has rank 5, the resulting matrix can have a maximum rank of 3.

5. Is matrix multiplication commutative?

No, matrix multiplication is not commutative. This means that the order in which matrices are multiplied matters. In general, AB is not equal to BA, except in special cases where one of the matrices is an identity matrix or a scalar multiple of the other matrix.

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