2.1 The Determinant of a Matrix · Linear Algebra Refresher

The determinant compresses an entire square matrix into one number whose only job, for our purposes, is to answer: singular or not? Cofactor expansion defines it; the real skill is choosing where to expand.

Small cases first, since you should never derive these on an exam:

det (a_{11}) = a_{11}, a c b d = a d - b c .

Definition · Minors and cofactors

For

n > 1

, let

M_{ij}

be the

(n - 1) \times (n - 1)

matrix obtained by deleting row $i$ and column $j$ of

A

. Its determinant

det (M_{ij})

is the minor of

a_{ij}

, and the cofactor is

A_{ij} = (- 1)^{i + j} det (M_{ij}) .

The sign pattern is a checkerboard starting with $+$ in the top-left corner. The determinant is then defined by expansion along the first row, $det (A) = a_{11} A_{11} + a_{12} A_{12} + \dots + a_{1 n} A_{1 n}$ .

Theorem 2.1.1 · Expand anywhere

For

n \geq 2

det (A)

can be computed by a cofactor expansion along any row or any column:

det (A) = j = 1 \sum n a_{ij} A_{ij} (any fixed i) = i = 1 \sum n a_{ij} A_{ij} (any fixed j) .

This freedom is the whole game: pick the row or column with the most zeros and most of the work disappears.

Theorem 2.1.2 & 2.1.3

det (A^{T}) = det (A)

for any square

A

; and if

A

is triangular (upper, lower, or diagonal),

det (A)

is the product of the diagonal entries.

If $A$ has a row or column of zeros, then $det (A) = 0$ (expand along it).
If $A$ has two identical rows or columns, then $det (A) = 0$ .
$det (I) = 1$ .

Worked example

Expanding along the emptiest column

0/4 steps

Evaluate the determinant with as little arithmetic as possible.

A = 0002241035010033

!Exam trap

The cofactor sign

(- 1)^{i + j}

is attached to the position, not to the entry. When you expand along row 2 of a

3 \times 3

, the signs run

-, +, -

. Forgetting this is the single most common determinant error.

Check your work

Determinant & inverse

Reduces to triangular form, tracking every sign flip. Square matrices only.

Square matrix

2	1	3
4	2	1
6	-3	4

det =-60

status

nonsingular

Ax = b

unique solution

A⁻¹

-11/60	13/60	1/12
1/6	1/6	-1/6
2/5	-1/5	0

Check yourself

Which row or column should you expand along to compute this determinant with the least work?

0002241035010033

T / F

det (A^{T}) = det (A)

for every square matrix

A

T / F

A

is triangular,

det (A)

is the product of its diagonal entries.

The cofactor

A_{32}

of a

4 \times 4

matrix equals

(- 1)^{3 + 2} det (M_{32})

. What is

M_{32}

2.1The Determinant of a Matrix

What you must be able to do

Expanding along the emptiest column

Determinant & inverse

Check yourself