Linear Algebra - Chapter 3 Determinants

Section 3.1 Introduction to Determinants

Definition:

• For $n = 2$ : $\det A = a_{11}a_{22}-a_{12}a_{21}$

• For $n \ge 2$, we have a recursive definition:

  $$\det A=\sum_{j=1}^n(-1)^{i+j}a_{1j}\det A_{1j}$$

  where $A_{1j}$ is a submatrix of $A$ that delete the first row and $j$ th column. (We called $\det A_{ij}$ a minor)

  Let:

  $$C_{ij}=(-1)^{ij}\det A_{ij}$$

  The $C_{ij}$ is called the $(i,j)$ - cofactor of $A$.

  Then

  $$\det A = a_{11}C_{11}+a_{12}C_{12}+\cdots +a_{1n}C_{1n}$$

  This formula is called a cofactor expansion across the first row of $A$

Theorem 1 :

The determinant of an $n\times n$ matrix $A$ can be computed by a cofactor expansion across any row or down any column.

The factor $(-1)^{i+j}$ determines the following checkerboard pattern of signs:

Theorem 2 :

If $A$ is a triangular matrix, then $\det A$ is the product of the entries on the main diagonal of $A$

Section 3.2 Properties of Determinants

Row Operations

Let $A$ be a square matrix

• If a multiple of one row of $A$ is added to another row to produce a matrix $B$, then $\det B =\det A$
• If two rows of $A$ are interchanged to produce $B$, then $\det B=-\det A$
• If one row of $A$ is multiplied by $k$ to produce $B$, then $\det B=k\det A$
  • Hence, we get $\det A=\dfrac 1 k \det B$ (Important!)

Invertible

If we use this row operations to get a echelon form $U$, If $U$ has $n$ pivots, then $\det U\neq 0$, and $\det A$ is invertible!

Theorem 4 :

A square matrix $A$ is invertible iff $\det A \neq 0$

Columns

If we change all the rows to columns, all our statement are still all right.

Theorem 5:

If $A$ is an $n\times n$ matrix, then $\det A^T=\det A$

Multiplicative Property

Theorem 6：

If $A$ and $B$ are $n\times n$ matrices, then $\det AB=(\det A)(\det B)$

But this not holds for summation!!! That is $\det(A+B)\neq \det A+\det B$

Linear Property

Let $A=[a_1\cdots a_{j-1}\;\;x\;\;a_{j+1}\cdots a_n]$, $T:\R \to \R^n$, then $T(x)=\det A$, we have:

• $cT(x)=T(cx)$
• $T(u+v)=T(u)+T(v)$

So $T(x)$ is a linear transformation.

A simple proof:

We can compute matrix $A$ by a cofactor expansion down the $j$-th column, and all the $\forall i :C_{i,j}$ are not changed, therefore:

$$\det A=\sum_{i}a_{ij}C_{ij}$$

So $T(x)$ is a linear transformation of $C_{ij}$.

Section 3.3 The Applications of Determinants

Cramer’s Rule

Let $A$ be an invertible $n\times n$ matrix. For any $b$ in $\R^n$, the unique solution $x$ of $Ax=b$ has entries given by

$$x_i=\frac{\det A_i(b)}{\det A}$$

The Inverse Formula

Let $A$ be an invertible $n\times n$ matrix, then:

$$A^{-1}=\frac 1 {\det A}\text{adj}\, A$$

The matrix of cofactors on the right side is called the adjugate (or classical adjoint) of $A$, denoted by $\text{adj}\, A$. (In fact, it is a transpose of the cofactor matrix)

Determinants as Area or Volume

• If $A$ is a $2\times 2$ matrix, the area of the parallelogram determined by the columns of $A$ is $|\det A|$. (Note, it is the area of parallelogram, not triangle)
• If $A$ is a $3\times 3$ matrix, the volume of the parallelepiped determined by the columns of $A$ is $|\det A|$.

Linear Transformations

Let $T:\R^2\to \R^2$ be the linear transformation determined by a $2\times 2$ matrix $A$. If $S$ is a parallelogram in $\R^2$, then

$$\{\text{area of } T(S)\} = |\det A|\cdot \{\text{area of } S\}$$

If $T$ is determined by a $3\times 3$ matrix $A$, and if $S$ is a parallelepiped in $\R^3$, then

$$\{\text{volume of } T(S)\} = |\det A|\cdot \{\text{volume of } S\}$$