03 Determinants

Question 1

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

C_ij = ___

Answer 1

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

C_ij = (-1)^i+j det A_ij

Question 2

The determinant of an n x n matrix

det A =

Answer 2

The determinant of an n x n matrix

det A = a_i1C_i1 + a_i2C_i2 + … + a_inC_in

Question 3

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

det A =

Answer 3

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)

Question 4

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

Answer 4

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

Question 5

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 = __

Answer 5

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

Question 6

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

Answer 6

If A is an n x n matrix, then det A^T = det A

Question 7

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

Answer 7

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

Question 8

Cramer’s Rule

Answer 8

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

A_i(b) = [a1 … b … a_n]

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

x_i = det A_i(b) / det A

Question 9

Let A be an invertible n x n matrix. Then

A^-1 =

Answer 9

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.

Question 10

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

Answer 10

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|.

Question 11

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)} = __

Answer 11

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}