Properties of Determinants Quiz

Properties of Determinants

• The determinant of the identity matrix equals ONE.
• If the absolute values of determinants are equal (|A| = |A'|), an even number of row swaps has occurred.
• If |A| = -|A'|, then an odd number of row swaps has happened.
• The determinant of a matrix | ta tb | | c d | transforms to t | a b | | c d |.
• The determinant | a + a' b + b' | | c d | can be expressed as | a b | | a' b' | + | c d |.
• The statement that det(A+B) equals det(A) + det(B) is FALSE.
• If two rows in a matrix are equal, the determinant is ZERO.
• A determinant of zero indicates that the matrix is NOT invertible.
• A determinant of zero confirms that the matrix is singular.
• Gaussian elimination does not change the value of the determinant (TRUE).
• A complete row of zeroes results in a determinant of ZERO.
• The matrix structured as [ d1 ... ] [ 0 d2 ... ] ... has a determinant that equals the product of the diagonal elements (d_i).
• The determinant detA = 0 implies singularity.
• If detA ≠ 0, this indicates the matrix is invertible.
• The determinant of the product of matrices (det(AB)) equals the product of their determinants (det(A) x det(B)).
• The BIG DETERMINANT FORMULA for an nxn matrix states the determinant equals the sum of n-factorial terms with alternating signs based on components.
• The COFACTOR FORMULA for the determinant expresses that for an nxn matrix, the determinant of A is calculated as a sum involving elements a_ij and their corresponding cofactors C_ij, applying the sign factor (-1)^(i+j).
• If matrix A is non-singular, then the determinant of its inverse is 1/det(A).