Are the following statements true or false?Answer 1 Question 10 If A is a square matrix with zero determinant, then one row of A must be a sum of two other rows.Answer 2 Question 10 Even if two columns of a square matrix are equal, the determinant of the matrix may be non-zero.Answer 3 Question 10 If any two rows of a square matrix are multiples of each other, then the determinant is always zero
Question
Are the following statements true or false?Answer 1 Question 10 If A is a square matrix with zero determinant, then one row of A must be a sum of two other rows.Answer 2 Question 10 Even if two columns of a square matrix are equal, the determinant of the matrix may be non-zero.Answer 3 Question 10 If any two rows of a square matrix are multiples of each other, then the determinant is always zero
Solution
Answer 1: False. A square matrix with a zero determinant does not necessarily mean that one row of A must be a sum of two other rows. A zero determinant indicates that the matrix is singular, not invertible, or its rows are linearly dependent. However, it doesn't necessarily mean that one row is the sum of two other rows.
Answer 2: False. If two columns of a square matrix are equal, the determinant of the matrix is always zero. This is because the determinant of a matrix with two identical rows or columns is zero, as it indicates that the rows or columns are linearly dependent.
Answer 3: True. If any two rows of a square matrix are multiples of each other, then the determinant is always zero. This is because the rows are linearly dependent, which means the matrix is singular and its determinant is zero.
Similar Questions
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