Determinants

Let Sn = {1, 2, . . . , n} be the set of integers from 1 to n. A

rearrangement of the elements in S is called a permutation of Sn.

The possible number of permutations Sn can have is n!, the factorial

of n, which is defined by

n! = n(n− 1)(n− 2) · · · 3 · 2 · 1.

Consider the permutation j1 j2 j3 . . . jr . . . js . . . jn. A permuta-

tion is said to be an inversion if a larger jr precedes a smaller jr. A

permutation is even (odd) if the total number of inversions is even

(odd).

Example: S3 = {(1 2 3), (1 3 2), (2 1 3), (2 3 1), (3 2 1), (3 1 2)}the odd permutations are: {(1 3 2), (3 2 1), (2 1 3)} and

the even permutations are : {(1 2 3), (3 1 2), (2 3 1)}.

Definition: Let A = [aij] be an n × n matrix. The determinant

of A, denoted by |A|, is given by

|A| =∑

ρ→n!εj1j2...jn a1j1 a2j2 . . . anjn

where

εj1j2...jn =

+1 if permutation is even

−1 if permutation is odd.

Examples:

1. If A is a 1× 1 matrix; i.e. A = [a11], then |A| = a11.

2. Determinant of order 2∣∣∣∣∣∣∣a11 a12

a21 a22

∣∣∣∣∣∣∣ = ε12a11a22 + ε21a21a12 = a11a22 − a21a12.

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3. Determinant of order 3

∣∣∣∣∣∣∣∣∣∣∣a11 a12 a13

a21 a22 a23

a31 a32 a33

∣∣∣∣∣∣∣∣∣∣∣=

ε123a11a22a33 + ε231a12a23a31 + ε321a13a22a31

+ ε213a12a21a33 + ε132a11a23a32 + ε312a13a21a32

= a11

∣∣∣∣∣∣∣a22 a23

a32 a33

∣∣∣∣∣∣∣ − a12

∣∣∣∣∣∣∣a21 a23

a31 a33

∣∣∣∣∣∣∣ + a13

∣∣∣∣∣∣∣a21 a22

a31 a32

∣∣∣∣∣∣∣ .

Exercises:

1.

∣∣∣∣∣∣∣2 3

−1 4

∣∣∣∣∣∣∣ 3.

∣∣∣∣∣∣∣∣∣∣∣1 0 6

3 4 15

5 6 21

∣∣∣∣∣∣∣∣∣∣∣5.

∣∣∣∣∣∣∣∣∣∣∣1 0 0

2 3 5

4 1 3

∣∣∣∣∣∣∣∣∣∣∣

2.

∣∣∣∣∣∣∣∣∣∣∣1 0 2

3 4 5

5 6 7

∣∣∣∣∣∣∣∣∣∣∣4.

∣∣∣∣∣∣∣∣∣∣∣2 3 5

1 0 1

2 1 0

∣∣∣∣∣∣∣∣∣∣∣

Properties of Determinants

1. |AT | = |A|.

2. If matrix B results from interchanging 2 rows (or columns) of

matrix A, |B| = −|A|.

3. If two rows (or columns) of A are equal, then |A| = 0.

4. If a row (or column) consists of entirely zero, then |A| = 0.

5. If matrix B results from multiplying a row of matrix A by a

scalar c, then |B| = c|A|.

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Cofactor Expansion

Definition: Let A = [aij] be an n × n matrix. Let Mij be the

(n − 1) × (n − 1) submatrix obtained by deleting the ith row and

jth column of A. The determinant of Mij is called the minor of

aij. The cofactor Aij of aij is

Aij = (−1)i+j|Mij|.

Theorem: Let A = [aij] be an n × n matrix. Then for each

1 ≤ i ≤ n, the cofactor expansion about the ith row

|A| = ai1Ai1 + ai2Ai2 + · · · + ainAin =n∑

k=1aikAik.

Also for 1 ≤ j ≤ n, the cofactor expansion about the jth column

|A| = a1jA1j + a2jA2j + · · · + anjAnj =n∑

k=1akjAkj.

Examples:

1.

∣∣∣∣∣∣∣∣∣∣∣2 1 −3

0 1 2

−4 2 1

∣∣∣∣∣∣∣∣∣∣∣2.

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 2 −3 4

−4 2 1 3

3 0 0 −3

2 0 −2 3

∣∣∣∣∣∣∣∣∣∣∣∣∣∣3.

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

2 2 −3 1

0 1 2 −1

3 −1 4 1

2 3 0 0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣APPLICATIONS:

Finding the inverse of a Matrix using determinants

Definition: Let A = [aij] be an n× n matrix. The adjoint of A,

denoted by adjA, is the n × n matrix whose ijth element is the

cofactor Aji of aij.

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Example: Compute for the adjoint of:

(a)

∣∣∣∣∣∣∣∣∣∣∣3 −2 1

5 6 2

1 0 −3

∣∣∣∣∣∣∣∣∣∣∣(b)

∣∣∣∣∣∣∣∣∣∣∣6 2 8

−3 4 1

4 −4 5

∣∣∣∣∣∣∣∣∣∣∣Theorem: If A = [aij] is an n× n matrix, then

A(adjA) = (adjA)A = |A|In.

Corollary: If A is an n× n matrix and |A| 6= 0, then

A−1 =1

|A|(adjA).

Theorem: A matrix A is nonsingular if and only if |A| 6= 0.

Corollary: AX = 0 has a non-trivial solution if and only if |A| = 0.

Examples: Determine whether the following matrix is non-singular:

1.

1 3 2

2 1 4

1 −7 2

2.

0 1 2

1 2 0

1 3 4

3.

1 2 0 5

3 4 1 7

−2 5 2 0

0 1 2 −7

Cramer’s Rule

Let AX = B be a linear system of n equations in n unknowns.

If |A| 6= 0 then if X = [x1 x2 x3 · · · xn]T then

xi =|Ai||A|

i = 1, 2, . . . , n,

where Ai is obtained by replacing the ith column of A by B.

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Exercise: Solve using Cramer’s Rule:

1.

2x + 4y + 6z = 2

x + 2z = 0

2x + 3y − z = −5

2.

2x + 3y + 7z = 2

−2x − 4z = 0

x + 2y + 4z = 0

3.

x + y + z − 2w = 2

2y + z + 3w = 4

2x + y − z + 2w = 5

x y + w = 4

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