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Questions and Answers
The determinant of the identity matrix equals
The determinant of the identity matrix equals
one
|A| = |A'| implies what number of row swaps?
|A| = |A'| implies what number of row swaps?
an even number
|A| = -|A'| implies what number of row swaps?
|A| = -|A'| implies what number of row swaps?
an odd number
What does | ta tb |
| c d | become?
What does | ta tb | | c d | become?
| a+a' b+b'|
| c d | becomes
| a+a' b+b'| | c d | becomes
Det(A+B) is equal to det(A) + det(B)
Det(A+B) is equal to det(A) + det(B)
If two rows are equal in the matrix, the determinant must be
If two rows are equal in the matrix, the determinant must be
A determinant of zero implies that the matrix is invertible
A determinant of zero implies that the matrix is invertible
A determinant of zero implies that the matrix is singular
A determinant of zero implies that the matrix is singular
Gaussian elimination does not affect the determinant
Gaussian elimination does not affect the determinant
A complete row of zeroes implies that the determinant of that matrix is
A complete row of zeroes implies that the determinant of that matrix is
[ d1 *************** ]
[ 0 d2 *********** ]
[ 0 0 d3 ****** ]
[ 0 0........** ]
[ 0 0.....0 dn ] has determinant:
[ d1 *************** ] [ 0 d2 *********** ] [ 0 0 d3 ****** ] [ 0 0........** ] [ 0 0.....0 dn ] has determinant:
DetA = 0 exactly when the determinant is
DetA = 0 exactly when the determinant is
DetA != 0 exactly when the determinant is
DetA != 0 exactly when the determinant is
Det(AB) =
Det(AB) =
BIG DETERMINANT FORMULA: for nxn, the determinant of a =
BIG DETERMINANT FORMULA: for nxn, the determinant of a =
COFACTOR FORMULA FOR DETERMINANT: for nxn the determinant of A =
COFACTOR FORMULA FOR DETERMINANT: for nxn the determinant of A =
If a is non-singular det(A inverse) =
If a is non-singular det(A inverse) =
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Study Notes
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).
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