Questions and Answers
The determinant of the identity matrix equals
one
|A| = |A'| implies what number of row swaps?
an even number
|A| = -|A'| implies what number of row swaps?
an odd number
What does | ta tb | | c d | become?
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| a+a' b+b'| | c d | becomes
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Det(A+B) is equal to det(A) + det(B)
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If two rows are equal in the matrix, the determinant must be
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A determinant of zero implies that the matrix is invertible
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A determinant of zero implies that the matrix is singular
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Gaussian elimination does not affect the determinant
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A complete row of zeroes implies that the determinant of that matrix is
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[ d1 *************** ] [ 0 d2 *********** ] [ 0 0 d3 ****** ] [ 0 0........** ] [ 0 0.....0 dn ] has determinant:
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DetA = 0 exactly when the determinant is
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DetA != 0 exactly when the determinant is
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Det(AB) =
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BIG DETERMINANT FORMULA: for nxn, the determinant of a =
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COFACTOR FORMULA FOR DETERMINANT: for nxn the determinant of A =
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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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Description
Test your understanding of the properties of determinants with this quiz. Explore key concepts such as the effects of row swaps, the implications of zero determinants, and more. Ideal for students studying linear algebra and matrix theory.
