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true or false. why? if a is an nxn matrix that is not invertible, then the matrix obtained by interchanging two rows of a cannot be invertible

1 Answer

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Answer:

True

the determinant is still zero

Explanation:

You want to know if it is true that if A is an n×n matrix that is not invertible, then the matrix obtained by interchanging two rows of a cannot be invertible, and why.

Invertible

The inverse of a matrix is the transpose of the cofactor matrix, divided by the determinant. The cofactor matrix of a square matrix can always be found, but division by the determinant fails if the determinant is zero.

The matrix is not invertible if the determinant is zero.

Row operations

Interchanging two rows of a matrix negates the determinant. If the determinant is zero, the opposite of the determinant is still zero.

A non-invertible matrix with rows interchanged remains non-invertible, because the determinant remains zero.

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