Determinant

🔗

Section 3.1 Determinant

🔗

We will

define a function on matrices,
discover an efficient algorithm for computing it, and
expand the Big Theorem incorporating this function.

🔗

Subsection 3.1.1 Definition of Determinant

🔗

For this course we will define the determinant as a calculation done recursively.

First we will define the determinant for $2 \times 2$ matrices.
Next we will define a recursive process for computing determinants of larger matrices.
For example the determinant of a $3 \times 3$ matrix will involve computing determinants of 3 $2 \times 2$ matrices.
Similarly the determinant of a $4 \times 4$ matrix will involve computing determinants of 4 $3 \times 3$ determinants which each involve computing 3 determinants of $2 \times 2$ matrices.
How many $2 \times 2$ determinants are required to compute the determinant of a $5 \times 5$ matrix?

🔗

Definition 3.1.1.

🔗

det (\begin{array}{rr} a & b \\ c & d \end{array}) = a d - b c .

Example 3.1.2.

det (\begin{array}{rr} 5 & 3 \\ - 1 & 7 \end{array}) = 5 (7) - (- 1) 3 = 38 .

Checkpoint 3.1.3.

Compute the determinant of

[\begin{array}{rr} 2 & 9 \\ 8 & 4 \end{array}]

🔗

Next we define a cofactor. This is just a signed determinant of a sub-matrix. It has no significance outside calculating determinants.

🔗

Definition 3.1.4.

🔗

The cofactor of position

i, j

is the determinant of the matrix after removing row

i

and column

j .

In the diagram the removed positions are in gray.

C_{i, j} = (- 1)^{i + j} det (\begin{array}{rrcrcr} a_{1, 1} & a_{1, 2}, & \dots & a_{1, j} & \dots & a_{1, n} \\ a_{2, 1} & a_{2, 2}, & \dots & a_{2, j} & \dots & a_{2, n} \\ ⋮ \\ a_{i, 1} & a_{i, 2}, & \dots & a_{i, j} & \dots & a_{i, n} \\ ⋮ \\ a_{m, 1} & a_{m, 2}, & \dots & a_{m, j} & \dots & a_{m, n} \end{array})

🔗

Note that the sign (

(- 1)^{i + j}

) of cofactors alternates in a checkerboard pattern. For a

4 \times 4

is looks like

\begin{array}{rrrr} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array}

🔗

Definition 3.1.5.

🔗

One way to compute a determinant is the following formula.

det (\begin{array}{rrcr} a_{1, 1} & a_{1, 2} & \dots & a_{1, n} \\ a_{2, 1} & a_{2, 2} & \dots & a_{2, n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m, 1} & a_{m, 2} & \dots & a_{m, n} \end{array}) = a_{1, 1} C_{1, 1} + a_{1, 2} C_{1, 2} + \dots + a_{1, n} C_{1, n}

🔗

Note that the elements of the first row are multiplied by their cofactors. One can use this formula with any row or any column.

Example 3.1.6.

Calculate the determinant.

\begin{aligned} det (\begin{array}{rrr} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 8 & 4 & 3 \end{array}) & = 1 (1 \cdot 3 - 4 \cdot 5) - 2 (1 \cdot 3 - 8 \cdot 5) + 3 (1 \cdot 4 - 8 \cdot 1) \\ = 1 (- 17) - 2 (37) + 3 (- 4) \\ = - 103 \end{aligned}

Checkpoint 3.1.7.

Based on this definition, how much fun do you expect it to be to calculate by hand a determinant of a

7 \times 7

matrix?

Example 3.1.8.

Calculate the determinant.

\begin{aligned} det (\begin{array}{rrr} 10 & 2 & 0 \\ 7 & 9 & - 5 \\ 3 & 6 & 0 \end{array}) & = 0 (stuff) - (- 5) (10 (6) - 3 (2)) + 0 (stuff) \\ = 5 (54) \\ = 270. \end{aligned}

🔗

Subsection 3.1.2 Efficient Algorithm

🔗

This activity will demonstrate the effect of changes to a matrix on the determinant. These effects can be used to more efficiently calculate determinants.

🔗

Activity 3.1.1.

🔗

Use the following matrices for this activity.

A = [\begin{array}{rrr} 1 & - 1 & 8 \\ - 7 & 1 & 9 \\ 3 & 1 & 4 \end{array}],

B = [\begin{array}{rrr} 1 & - 1 & 8 \\ - 7 & 1 & 9 \\ 6 & 2 & 8 \end{array}],

C = [\begin{array}{rrr} 1 & - 1 & 8 \\ 3 & 1 & 4 \\ - 7 & 1 & 9 \end{array}],

D = [\begin{array}{rrr} 1 & - 1 & 8 \\ - 5 & - 1 & 25 \\ 3 & 1 & 4 \end{array}] .

🔗

Note

$B$ is the result of the row operation $R_{3} \leftarrow 2 R_{3}$ applied to matrix $A$
$C$ is the result of the row operation $R_{2} \leftrightarrow R_{3}$ applied to matrix $A$
$D$ is the result of the row operation $R_{2} \leftarrow 2 R_{1} + R_{2}$ applied to matrix $A$