10 Transpose of a matrix

Transpose of a matrix

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Let's define a transpose matrix.

Transpose of a transpose matrix A is same with the matrix A!

Determinant of transpose

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Let's assume determinant of any nxn matrix B is equal to the determinant of B's transpose.

Then will it be the same if the matrix is (n+1) x (n+1)?

Transpose of sums and inverses

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This means that C transpose is equal to the transpose of (A+B)!

Rank(a) = rank(transposse of a)

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The rank of matrix A is equal to the rank of its transpose.

The rank of A transpose is equal to the dimension of column space of A transpose. This is equal the number of basis vectors for the column space of A transpose.

Same applies to A.

Showing that A-transpose x A is invertible

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Let's study a transpose times A.

A transpose A will be a square matrix.

To show that is invertible, we have to prove all the columns are independent.

And if you get a k by k identity matrix, it means that your matrix is invertible.

This is the same thing with Av dot Av.

So we know Av must be equal to zero.

If v is a member of the nullspace of A transpose A & v is also a member of the nullspace of A, It only contains the zero vector!

This means that the only solution is the x is equal to the zero vector. That means that the column s of A tranpose A are linearly independent!!