Determinants

DETERMINANTS

$\begin{displaymath}\mbox{determinant of $\left(\begin{array}{cc} a&b\\ c&d\\ \... ...begin{array}{cc} a&b\\ c&d\\ \end{array}\right\vert = ad -bc.\end{displaymath}$

Properties of the Determinant
1. Any matrix A and its transpose have the same determinant, meaning

det A = det A^T

This implies that whenever we use rows, a similar behavior will result if we use columns.

2. The determinant of a triangular matrix is the product of the elements of the diagonal, that is

$\begin{displaymath}\left\vert\begin{array}{cc} a&b\\ 0&d\\ \end{array}\right\v... ...rt\begin{array}{cc} a&0\\ b&d\\ \end{array}\right\vert = ad .\end{displaymath}$

3. If we interchange even number of rows, the determinant of the new matrix is the negatieve of the old one, that is

$\begin{displaymath}\left\vert\begin{array}{cc} a&b\\ c&d\\ \end{array}\right\v... ...left\vert\begin{array}{cc} c&d\\ a&b\\ \end{array}\right\vert\end{displaymath}$

in this case we have interchanged two rows

4. If we multiply one row with a constant, the determinant of the new matrix is the determinant of the old one multiplied by the constant, that is

$\begin{displaymath}\left\vert\begin{array}{cc} \lambda a&\lambda b\\ c&d\\ \en... ...rray}{cc} a&b\\ \lambda c&\lambda d\\ \end{array}\right\vert.\end{displaymath}$

5. If we add one row to another one multiplied by a constant, the determinant of the new matrix is the same as the old one, that is

$\begin{displaymath}\left\vert\begin{array}{cc} a + \lambda c&b + \lambda d\\ c&... ...} a&b\\ c + \lambda a&d + \lambda b\\ \end{array}\right\vert.\end{displaymath}$

6. We have

det (AB) = det (A)det(B)

and

$\begin{displaymath}\det(A^{-1}) = \frac{1}{\det(A)}.\end{displaymath}$

Application of Determinant to Systems: Cramer’s Rule

We can use determinants and its properties in solving linear systems for which the matrix coefficient is nonsingular (or invertible).

Consider the linear system (in matrix form)
AX=B

where A is the matrix coefficient, B the non-homogeneous term, and X the unknown column-matrix.

We have:
Theorem. The linear system AX = B has a unique solution if and only if A is invertible. In this case, the solution is given by the so-called Cramer’s formulas:

$\begin{displaymath}x_i = \frac{\det(A_i)}{detA}\;,\;\; \mbox{for $i=1,\cdots,n$}\end{displaymath}$

where x_i are the unknowns of the system or the entries of X, and the matrix A_i is obtained from A by replacing the i^th column by the column B. In other words, we have

$\begin{displaymath}x_i = \frac{b_1 A_{1i} + b_2 A_{2i} + \cdots + b_n A_{ni}}{\det(A)}\end{displaymath}$

where the b_i are the entries of B.
thus, if the linear system AX = B is homogeneous, meaning B=0, and if A is invertible, the only solution is the trivial one, that is X=0. Hence if we are looking for a nonzero solution to the system, the matrix coefficient A must be singular or noninvertible. We also know that this will happen if and only if det(A)=0.

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Determinants

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