Determinant

Calculation

Determinant of 2x2 matrix

$$ A= \begin{bmatrix} a & b \newline c & d \end{bmatrix} $$

$|A| = det(A) = ad -bc$

Determinant of 3x3 matrix, also called expansion of the determinant by first row. Link.

$$ B= \begin{bmatrix} a & b & c \newline d & e & f \newline g & h & k \end{bmatrix} $$

$|B| = det(B) = a\begin{vmatrix} e & f \newline h & k \end{vmatrix} -b\begin{vmatrix} d & f \newline g & k \end{vmatrix} +c\begin{vmatrix} d & e \newline g & h \end{vmatrix}$

Or we can replicate first two columns, we got $$ \begin{bmatrix} a & b & c & a & b\newline d & e & f & d & e\newline g & h & k & g & h \end{bmatrix} $$

$|B| = (aek + bfg + cdh) - (bdk+afh+ceg)$, which is adding the sum of products of diagonals from top left to bottom right minus sum of products of diagonals from top right to bottom left.

Singular (degenerate) Matrix

if det(A) = 0, we call A is singular.

Rank

The rank of a matrix A is the dimension of the space generated (or spanned) by its columns, which is maximal number of linearly independent columns of A (or rows). Rank is thus a measure of the “nondegenerateness” of the system of linear equations.

Full rank

If matrix’s rank = min(rows, columns)

Rank Deficient:

If matrix is not full rank.

For example:

\begin{bmatrix} 1 & 0 & 1 \newline -2 & -3 & 1 \newline 3 & 3 & 0 \end{bmatrix}

Rank = 2, since the last column 2 = column 0 - columm 1

$$ A = \begin{bmatrix} 1 & 1 & 0 & 2 \newline -1 & -1 & 0 &-2 \end{bmatrix} $$

Rank(A) = 1

$$A^T= \begin{bmatrix} 1 & -1 \newline 1 & -1 \newline 0 & 0 \newline 2 & -2 \end{bmatrix} $$

$Rank(A^T) = 1$, $Rank(A) = Rank(A^T)$,