03 Determinants Flashcards Preview

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Flashcards in 03 Determinants Deck (11)
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1
Q

The (i, j)-cofactor of A is the number Cij given by

Cij = ___

A

The (i, j)-cofactor of A is the number Cij given by

Cij = (-1)i+j det Aij

2
Q

The determinant of an n x n matrix

det A =

A

The determinant of an n x n matrix

det A = ai1Ci1 + ai2Ci2 + … + ainCin

3
Q

Suppose a square matrix A has been reduced to an echelon form U by row replacements and row interchanges.

det A =

A

Suppose a square matrix A has been reduced to an echelon form U by row replacements and row interchanges.

det A = (-1)r * product of pivots in U
(if A is invertible)

4
Q

If A is a triangular matrix, then det A is ____

A

If A is a triangular matrix, then det A is the product of the entries on the main diagonal of A.

5
Q

Let A be a square matrix

  • If a multiple of one row of A is added to another row to produce a matrix B, then det B = __
  • If two rows of A are interchanged to produce B, then det B = __
  • If one row of A is multiplied by k to produce B, then det B = __
A

Let A be a square matrix

  • If a multiple of one row of A is added to another row to produce a matrix B, then det B = det A
  • If two rows of A are interchanged to produce B, then det B = - det A.
  • If one row of A is multiplied by k to produce B, then det B = k det A
6
Q

If A is an n x n matrix, then det AT = ___

A

If A is an n x n matrix, then det AT = det A

7
Q

If A and B are n x n matrices, then det AB = ___

A

If A and B are n x n matrices, then det AB = (det A)(det B)

8
Q

Cramer’s Rule

A

For any n x n matrix A and any b in Rn, let Ai(b) be the matrix obtained from A by replacing column i by the vector b.

Ai(b) = [a1 … b … an]

Cramer’s Rule

Let A be an invertible n x n matrix. For any b in Rn, the unique solution x of Ac = b has entries given by

xi = det Ai(b) / det A

9
Q

Let A be an invertible n x n matrix. Then

A-1 =

A

Let A be an invertible n x n matrix. Then

A-1 = (1/det A) adj A

where adj A is the transpose of the matrix of cofactors.

10
Q

If A is a 2 x 2 matrix ____ is |det A|. If A is a 3 x 3 matrix, ______ is |det A|.

A

If A is a 2 x 2 matrix the area of the parallelogram determined by the columns of A is |det A|. If A is a 3 x 3 matrix, the volume of the parallelepiped determined by the columns of A is |det A|.

11
Q

Let T : R2 -> R2 be the linear transformation determined by a 2 x 2 matrix A. If S i a parallelogram in R2, then

{area of T(S)} = __

If T is determined by a 3 x 3 matrix A, and if S is parallelepiped in R2, then

{volume of T(S)} = __

A

Let T : R2 -> R2 be the linear transformation determined by a 2 x 2 matrix A. If S i a parallelogram in R2, then

{area of T(S)} = |det A| {area of S}

If T is determined by a 3 x 3 matrix A, and if S is parallelepiped in R2, then

{volume of T(S)} = |det A| {volume of S}