Determinant

Determinant is a number associated with each matrix. The number can tell a lot about the nature of the matrix.

Some uses of determinants are as follows

• Test for invertibility: If the determinant of a matrix $A$ is zero, then $A$ is singular and not invertible.
• The determinant of $A$ equals the volume os a box in n-dimensional space.
• The determinant gives a formula for each pivot. We can decide the step when a rwo exchange is required.

Properties of Determinant

1. The determinant of identity matrix is 1
2. The determinant changes sign when two rows or columns are exchanged.
3. If two rows are equal then $det(A) = 0$

The whole point of linear algebra is to solve a system of linear equations and analyze the space they are in. The new methods introduced in the field are to solve the systems efficiently. Matrix notation is chosen to represent the system of linear equations in n-dimensions. The $n$ unknowns are written as vector $x$.

Linear Equations

$$ Ax = b $$

For example,

$$ 2x_1+3x_2-x_3 = 5 \\ 3x_1-4x_2+5x_3 = 9 \\ x_1-x_2+4x_3 = 5 $$

is a system of linear equations with three variables or three unknowns. The system is represented using matrices as:

$$ \begin{bmatrix} 2&3&-1 \\ 3&-4&5 \\ 1&-1&4 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 5 \\ 9 \\ 5 \end{bmatrix} $$

The meaning of solving the system of linear equations is to find the points of intersection. If the system has a single point of intersection then we say the system has a solution.

In general,

$$ \begin{bmatrix} a_{11} &\dots&a_{1n} \\ \vdots&&\vdots \\ a_{m1}&\dots&a_{mn} \end{bmatrix} \begin{bmatrix}x_1\\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix}b_1\\ \vdots \\ b_m\end{bmatrix} $$