Lesson 3.1 - The Determinant of a Matrix

When expanding by cofactors, zero entries have cofactors of what?

0, since a_ij C_ij = (0)C_ij = 0

What is the cofactor C_ij of the entry a_ij equal to?

C_ij = (-1)^i+j M_ij

How can you find the determinant of a 3x3 matrix, using expansion by cofactors?

Copy the 1st & 2nd columns of A to form 4th & 5th columns Then, add/subtract the products of the six diagonals

How can the determinant, larger than a 2x2, of a matrix be found?

Evaluating by expanding in any row or column

Definition of the Determinant of a 2 x 2 Matrix

The determinant of a 2 x 2 matrix, A, is det(A) = |A| = ad - bc

In regards to triangular matrices, what is a diagonal matrix?

a matrix in which all entries above/below the main diagonal are zero

What denotes the "determinant of A"?

det(A) or |A|

According to cofactor/minor +'s and -'s patterns, what type of signs do even positions have?

even positions (i + j is even) have positive signs

How do you obtain the cofactors of a matrix, using +'s and -'s signs?

find the minors & then apply the +'s and -'s patterns

Definition of the Determinant of a Square Matrix

if A is a square matrix of order n ≥ 2, then the determinant of A is the sum of the entries in the first row of A multiplied by their respective cofactors det(A) = |A| = ∑ a_1j C_1j + a_12 C_12 + ... + a_1n C_1n

Minors and Cofactors of Square Matrix

if A is a square matrix, then the minor M_ij of the entry a_ij is the determinant of the matrix obtained by deleting the ith row & jth column of A

Determinant of a Triangular Matrix

if A is a triangular matrix of order n, then its determinant is the porduct of the entries on the main diagonal: det(A) = |A| = a_11 a_22 a_33 ... a_nn

According to cofactor/minor +'s and -'s patterns, what type of signs do odd positions have?

odd positions (i + j is odd) have negative signs

What three possibilities can the determinant of a matrix be?

positive, zero, or negative

When expanding by cofactors, which row/column is usually the best choice for expansion?

row/column containing the most zeros

What do minors and cofactors differ in?

signs

What is the determinant of a matrix of order 1?

the entry of the matrix