Determinants

Idea. Use expansion along the first row (signs ). Compute three minors.
Step 1. Minors.
Step 2. Combine with signs. Zero determinant also hints that the third row is the sum of the first two, so rows are dependent.

Idea. Again expand along the first row, or notice a quick pattern: rows dependent determinant . We also verify by expansion.
Step (verification).

Idea. For a matrix . Use reciprocal trig identities , .
Step.

We expand the determinant: Subtract the first row from the second and third rows to simplify: giving Now expand along the first row: Calculate each minor: Therefore,

We can use properties of determinants or expand along the first row: Subtract the first column from the second and third columns: Now, Recall the factorization: Thus, Factor out and from rows: Calculate the determinant: Rewrite inside: So, Note the order: To match the common Vandermonde pattern, reorder factors and account for sign changes: \ Final answer:

We are given the matrix: To evaluate the determinant, we can expand along the first row: Now calculate each 2x2 determinant: Substituting back: Thus, the determinant of the matrix is .

We are given the matrix: To evaluate the determinant, we expand along the first row: Now, calculate each 2x2 determinant: Substitute these values back: Thus, the determinant is .

We are given the matrix: To evaluate the determinant of this 3x3 matrix, we use the cofactor expansion along the third row (because it contains two zeros, which simplifies the calculation). The cofactor expansion is: Since all terms contain a factor of 0, the determinant is: Thus, the correct answer is: (A) 0.

We are given the matrix: To find the determinant of a 3x3 matrix, we use the formula: For the given matrix: The determinant calculation becomes: After performing the multiplication and subtraction, we find that the determinant is . Thus, the correct answer is 0.

We are given the matrix: The determinant of a 2x2 matrix is calculated as: For the given matrix: The determinant is: Thus, the correct answer is .

We are given the matrix equation: The determinant of this 2x2 matrix is: Set the determinant equal to 0: Solve for : Thus, the correct answer is .