Matrix Properties

Trace

Trace of a matrix is the sum of its diagonal elements:

$$Tr(A) = \sum_iA_{ii}$$

Further,

$$Tr(A+B) = Tr(A) + Tr(B) \\ Tr(AB) = Tr(BA), \hspace{0.1cm}even\hspace{0.1cm}if\hspace{0.1cm}AB \neq BA \\ Tr(A) = Tr(A^T)$$

​Determinant

For a matrix, $A = \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}$​, it's determinant is given by:

$$|A| = a_{11}a_{22} - a_{12}a_{22}$$

For a $3 \times 3$ matrix, $A = \begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i \end{bmatrix}$​, it's determinant is given by:

$$|A| = a(ei-fh) - b(di-fg) + c(dh - eg)$$

​For a $4 \times 4$​ matrix, the determinant is given by:

$$|A| = \begin{bmatrix} a \times \begin{vmatrix} f & g & h\\ j & k & l\\ n & o & p \end{vmatrix} \end{bmatrix} - \begin{bmatrix} b \times \begin{vmatrix} e & g & h\\ i & k & l\\ m & o & p \end{vmatrix} \end{bmatrix} + \begin{bmatrix} c \times \begin{vmatrix} e & f & h\\ i & j & l\\ m & n & p \end{vmatrix} \end{bmatrix} - \begin{bmatrix} d \times \begin{vmatrix} e & f & g\\ I& j & k\\ m & n & o \end{vmatrix} \end{bmatrix}$$

Physically, determinant represents volume formed by column vectors.

Invertibility

A square matrix $A$ is invertible if and only if $det(A) \neq 0$. This automatically means that columns of $A$​ are linearly independent. For example,

$$A = \begin{bmatrix} v_1 & v_2 & ... & v_n\\ \end{bmatrix} \\ v_1 = (a_{11}, a_{21}, ... , a_{n1}) \\ v_2 = (a_{12}, a_{22}, ... , a_{n2}) \\ v_n = (a_{1n}, a_{2n}, ... , a_{nn})$$

​Let's presume the following:

$$v_n = \alpha_1v_1 + \alpha_2v_2 + ... + \alpha_{n-1}v_{n-1}$$

​In such a scenario, the area formed by the (hyper)parallelogram by the "n vectors" (the determinant) would be zero.

Another simpler example, consider a $2 \times 2$​ matrix where $v_1 = 2v_2$​. In such a case, the area of the parallelogram is zero.