Mathematical Concepts Of Matrix

Identity matrix

$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ is a $3 \times 3$ identity matrix. Identity matrix must be square matrix.

Rank

number of linearly independent rows (or columns). It is proved that numbers of independent rows and columns are the same.

To find the rank, transform the matrix into echelon form

(1)
\begin{bmatrix} 0 & -11 & -4 \\ 2 & 6 & 2 \\ 4 & 1 & 0 \end{bmatrix} \Rightarrow \begin{bmatrix} 4 & 1 & 0 \\ 2 & 6 & 2 \\ 0 & -11 & -4 \end{bmatrix} \Rightarrow \begin{bmatrix} 1 & \frac{1}{4} & 0 \\ 2 & 6 & 2 \\ 0 & -11 & -4 \end{bmatrix} \Rightarrow \begin{bmatrix} 1 & \frac{1}{4} & 0 \\ 0 & \frac{11}{2} & 2 \\ 0 & -11 & -4 \end{bmatrix} \Rightarrow \begin{bmatrix} 1 & \frac{1}{4} & 0 \\ 0 & 1 & \frac{4}{11} \\ 0 & 0 & 0 \end{bmatrix}

the rank is 2.

Transpose

(2)
\begin{align} A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \end{align}

transpose of A

(3)
\begin{align} A'=\begin{bmatrix} 1 & 4\\ 2 & 5 \\ 3 & 6 \end{bmatrix} \end{align}

matrix.svg

Singular

no pair. $A \cdot {A}^{-1}=I$, it's pair, so A is non-singular.

Determinant

|A| denotes determinant of A.

(4)
\begin{align} |A|= \begin{vmatrix} {a}_{11} & {a}_{12} \\ {a}_{21} & {a}_{22} \end{vmatrix}={a}_{11}{a}_{22}-{a}_{21}{a}_{12} \end{align}

Determinant=0, matrix is singular.

(5)
\begin{vmatrix} {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33} \end{vmatrix}= {a}_{11}\begin{vmatrix} {a}_{22} & {a}_{23} \\ {a}_{32} & {a}_{33}\end{vmatrix} -{a}_{12}\begin{vmatrix} {a}_{21} & {a}_{23} \\ {a}_{31} & {a}_{32}\end{vmatrix} +{a}_{13}\begin{vmatrix} {a}_{21} & {a}_{22} \\ {a}_{31} & {a}_{32} \end{vmatrix}

For matrix whose rank larger than 4, we can only use method above.

For $3 \times 3$ matrix,also, like $2 \times 2$ matrix, determinant = left-up to right-down diagnose $-$ left-down to right-up diagonose=(a11a22a33+a12a23a31+a21a32a13)-a21a12a33-a31a22a13-a32a23a11

Inverse

(6)
\begin{align} {A}^{-1}=\frac{1}{|A|}C' \end{align}

where

(7)
\begin{align} C=\begin{bmatrix} |{C}_{11}| & |{C}_{12}| \\ |{C}_{21}| & |{C}_{22}| \end{bmatrix} \end{align}
(8)
\begin{align} |{C}_{12}|=-\begin{vmatrix} {a}_{21} & {a}_{23} \\ {a}_{31} & {a}_{32}\end{vmatrix} \end{align}

Cramer's Rule

(9)
\begin{align} {{x}_{j}}^{*}=\frac{|{A}_{j}|}{|A|}=\frac{1}{|A|} \cdot \begin{vmatrix} {a}_{11} & {a}_{12} & \cdots & {d}_{1} & \cdots & {a}_{1n}\\ {a}_{21} & {a}_{22} & \cdots & {d}_{2} & \cdots & {a}_{2n}\\ \vdots & \vdots & & \vdots & & \vdots\\ {a}_{n1} & {a}_{n2} & \cdots & {d}_{n} & \cdots & {a}_{nn}\\ \end{vmatrix} \end{align}

column d locates in jth column.

See also

Matrix

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