Matrix Determinants and Properties

The video tutorial explains the concept of an n by n matrix, focusing on row vectors and their representation. It discusses the process of swapping rows in a matrix and how this affects the determinant. The tutorial highlights that if two rows are identical, the determinant becomes zero, indicating non-invertibility. This is linked to the concept of reduced row echelon form and matrix invertibility.

What is the primary characteristic of an n by n matrix as introduced in the video?

• It is always invertible.

• It is always singular.

• It has n rows and m columns.

• It has n rows and n columns.

How can a matrix be rewritten using row vectors?

• By using diagonal vectors.

• By using scalar multiplication.

• By using row vectors for each row.

• By using column vectors.

What happens to the determinant when two rows of a matrix are swapped?

• It doubles.

• It remains the same.

• It becomes the negative of the original determinant.

• It becomes zero.

What is the result of swapping two identical rows in a matrix?

• The matrix becomes singular.

• The matrix remains unchanged.

• The determinant becomes positive.

• The matrix becomes invertible.

If a matrix has two identical rows, what is the determinant?

• It is undefined.

• It is zero.

• It is negative.

• It is positive.

Why does having duplicate rows in a matrix lead to a zero determinant?

• Because it makes the matrix invertible.

• Because it creates a row of zeros in row reduction.

• Because it changes the matrix to a diagonal form.

• Because it increases the rank of the matrix.

What is the relationship between duplicate rows and invertibility?

• Duplicate rows make a matrix non-invertible.

• Duplicate rows increase the determinant.

• Duplicate rows do not affect invertibility.

• Duplicate rows make a matrix invertible.

What is a key takeaway regarding duplicate rows and the determinant?

• Duplicate rows make the determinant positive.

• Duplicate rows have no effect on the determinant.

• Duplicate rows increase the determinant.

• Duplicate rows result in a zero determinant.

What happens to the determinant if a matrix has duplicate columns?

• It becomes positive.

• It becomes negative.

• It becomes zero.

• It remains unchanged.

What is the implication of having rows that are linear combinations of other rows?

• The determinant becomes positive.

• The matrix becomes invertible.

• The determinant becomes zero.

• The matrix becomes singular.