Chapter 2: Determinants (2.1 - 2.3)

Question 1

Define

The minor of entry a_ij (denoted M_ij)

Answer 1

The determinant of the submatrix that remains after the i_th row and j_th column are deleted from A.

Question 2

The cofactor of entry a_ij (denoted C_ij)

Answer 2

The number (-1)^i+j M_ij

Question 3

Computation of the determinant of an nxn matrix A using cofactors

Answer 3

Multiply the entries in any row (or column) by their cofactors, and add the resulting products.

i.e.: det(A) = a_1jC_1j + a_2jC_{2j + … +}a_njC_nj

for any 1 ≤ j ≤ n

Question 4

det(A), where A is an nxn triangular matrix

Answer 4

det(A) is the product of the of the entries on the main diagonal:

det(A) = a₁₁a₂₂ … a_nn

Question 5

The determinant of a square matrix with a row/column of 0s

Answer 5

0

Question 7

det(A^T)

Answer 7

det(A^T) = det(A)

Question 9

det(B), when B is a result of a single row/column of A multiplied by a scalar k

A&B = nxn matrices

Answer 9

det(B) = k.det(A)

Question 11

det(B), when B is a result of two rows/columns of A being interchanged.

A&B = nxn matrices

Answer 11

det(B) = - det(A)

Question 13

det(B), when B is the result of a multiple of one row/column of A being added to another row/column.

A&B = nxn matrices

Answer 13

det(B) = det(A)

Question 15

det(E), if E results from multiplying a row of I_n by k

Answer 15

det(E) = k

Question 17

det(E), if E results from interchanging two rows of I_n

Answer 17

det(E) = -1

Question 19

det(E), if E results from adding a multiple of one row of I_n to another

Answer 19

det(E) = 1

Question 21

det(A), A is a square matrix with two proportional rows / columns

Answer 21

det(A) = 0

Question 23

det(kA), where A is an nxn matrix

Answer 23

kⁿ.det(A)

Question 25

det(EB), where E & B are square matrices of the same size

Answer 25

det(EB) = det(E).det(B)

Question 27

How do we use determinants to determine whether a matrix is invertible

Answer 27

A is invertible iff (if and only if) det(A) ≠ 0

Question 29

det(A^-1)

Answer 29

det(A^-1) = ¹/_{det(<strong>A</strong>)}

Question 31

adj(A), a.k.a. adjoint of A

Answer 31

The transpose of the matrix of cofactors from A.

Question 33

Give A^-1, using adj(A) when A is an invertible matrix

Answer 33

A^-1 = ¹/_{det(<strong>A</strong>)} . adj(A)

Question 35

Cramer’s Rule

Answer 35

If Ax = b is a system of linear equations in n unknowns:

x₁ = ^{det(<strong>A1</strong>)}/_{det(<strong>A</strong>)}, x₂ = ^{det(<strong>A2</strong>)}/_{det(<strong>A</strong>)}, …

where A_j is the matrix obtained by replacing the entries in the j^th column of A by:

b = [b₁; b₂; … b_n] (b is a column matrix, not a row matrix)

Question 37

7 Equivalent statements if A is an nxn matrix:

Answer 37

• A is invertible
• Ax = 0 has only the trivial solution
• The reduced row echelon form of A is I_n
• A can be expressed as a product of elementary matrices
• Ax = b is consistent for every nx1 matrix b
• Ax = b has exactly one solution for every nx1 matrix b
• det(A) ≠ 0