Inverse Matrix

Identity matrix. An $n\times n$ matrix (square matrix) of the form

$\displaystyle I_n = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \hdotsfor{4} \\ 0 & 0 & \cdots & 1 \end{bmatrix}$

is called the $n\times n$ identity matrix. We simply write

when the size is clear from the content. It satisfies

$\displaystyle I A = A I = A$

for any square matrix

of the same size.

Inverse Matrix. Let be an $n\times n$ matrix. If there exists a matrix of the same size satisfying

$\displaystyle A C = I$ and $\displaystyle C A = I,$

then

is said to be invertible, and

is called the inverse of

, denoted by $A^{-1}$ . If

is not invertible, we call

a singular matrix.

Matlab/Octave. The function eye(n) generates the $n\times n$ identity matrix. For example,

> eye(5)

creates the $5\times 5$ identity matrix.

EXAMPLE 1 Find $3\times 3$ and $5\times 5$ identity matrix.

EXAMPLE 2. Let $A = \left[\begin{array}{cc} 2 & 5 \\ -3 &-7 \end{array}\right]$ and $C = \left[\begin{array}{cc} -7 &-5 \\ 3 & 2 \end{array}\right]$ . Then show that $C = A^{-1}$ .