# Introduction to Linear Algebra

Recall we have previously defined the determinant of a 2 ×
2 matrix

as How
was this use in the

formula for the inverse of a 2 × 2 matrix?

The determinant of any matrix A is usually denoted by either |A| or detA.

We may also write

We essentially replace the bracket notation [ ] around the matrix with

absolute value type notation | |, indicated we want the determinant of the

matrix.

We define the determinant of a 1 × 1 matrix A as

where |a| does NOT indicate absolute value.

## Determinant of a 3 × 3 Matrix

The determinant of a 3 × 3 matrix can be thought of as a linear combination

of determinants of 2 × 2 matrices. The process is exactly the same as the

cross product .

Then the determinant of A is the scalar

We define the (i, j) minor of A, A

_{ij }, as the submatrix of A obtained by

deleting row i and column j. We may then write

Ex: Find the determinant of

## Determinant of an n × n Matrix

Let A = [a

_{ij}] be an n × n matrix, n ≥ 2. Then the determinant of A is

The minor with its sign , is the (i, j)− cofactor of A :

Then we may write

Interestingly, the determinant can be computed similarly using any other row.

Laplace Extension Thm: An n × n matrix A = [a

_{ij}], n ≥ 2 has

These are the cofactor expansion along the i

^{th }row and j

^{th}column,

respectively

## Computing determinants

Ex: The previous theorem tells us that we may choose any row or column to

compute the determinant along. Use this idea to find the determinant of

each of the following:

## Properties of determinants

Thm: Let A = [A

_{ij}] be a square matrix.

(a) If A has a zero row (column), then detA = 0.

(b) If B is obtained by interchanging two rows (columns) of A, then

detB = −detA.

(c) If A has two identical rows (columns), then detA = 0.

(d) If B is obtained by multiplying a row (column) of A by k, then

detB = l detA.

(e) If A,B and C are identical except that the i

^{th }row (column) of C is the

sum of the i

^{th}rows (columns) of A and B, then detC = detA + detB.

(f) If B is obtained by adding a multiple of one row (column) of A to

another row (column), then detB = detA.

Ex: Compute detA for

Ex: Compute detA forby
reducing to rref . Remember to keep

track of scalars!

## Determinants of Elementary Matrices

Thm: Let E be an n × n elementary matrix.

(a) If E results from interchanging two rows of I

_{n}, then detE = −1.

(b) If E results from multiplying one row or I

_{n}by k, then detE = k.

(c) If E results from adding a multiple of one row of I

_{n}to another row, then

detE = I.

Lemma: Let B be an n × n matrix and E be an n × n elementary matrix.

Then

The previous theorem and lemma can be used to prove the following

Thm: A square matrix A is invertible iff detA ≠ 0.

Pf: The idea is the you reduce A to rref R using elementary matrices

E_{1}, . . . ,E_{r}. Write

and take the determinant of both sides.

## Determinants and Matrix Operations

Thm: If A is an n × n matrix, then det (kA) = k

^{n}detA.

Note: In general, there is no formula for det (A + B).

Thm: If A and B are n × n matrices, then

Ex: For A =

find det AB by first calculating AB and finding its determinant, and then by using the theorem.

Thm: If A is invertible, then

Ex: Find det (A^{−1}) for A from the previous exercise by
explicitly finding A^{−1}

and by using the previous theorem.

## Aside: Cramer’s Rule and the Adjoint

The book presents Cramer’s Rule, which allows you to solve small matrix

problems easily. The idea is that we can find the solution to a linear system

simply from computing determinants.

The text also presents the adjoint of a matrix. These can both be used to

help prove the Laplace Expansion Theorem. Their practical application is

limited, thus we will discuss these in class only if time allows. If you are a

Math major, make sure you read through each of these concepts.

The last theorem for this chapter is used as part of the proof of the Laplace

Expansion Theorem:

Thm: Let A be an n × n matrix and let B be obtained by interchanging any

two rows (columns) of A. Then

## Matlab Determinants

As with most subjects in linear algebra , Matlab has a very intuitive way to

compute the determinant. We simply define a matrix, and apply the function

det. To find the determinant of

we do:

» A = [3,1,2;3,1,0;0,1,4];

» det(A)

Let us say we wish to verify det (A

^{10})= (detA)

^{10}. We do:

» A = [3,1,2;3,1,0;0,1,4];

» det(A^10)

» det(A)^10

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