Chapter 2, Problem 18RP

Explanation of Solution

Determinant of the given matrix:

Consider the following matrix,

$A=\left[\begin{array}{l}\text{2 4 6}\\ \text{1 0 0}\\ \text{0 0 1}\end{array}\right]$

The  $\det A$ is defined as follows:

$\det A={(-1)}^{i+1}{a}_{i1}\left(\det A_{i1}\right)+{(-1)}^{i+2}{a}_{i2}\left(\det A_{i2}\right)+\cdots +{(-1)}^{i+m}{a}_{im}\left(\det A_{im}\right)$

$i=1,2,\dots ,m$

From the matrix $A$, the cofactor matrices are determined as follows:

Here,

$a_{11}=2,a_{12}=4,\text{ and }{a}_{13}=6$

The first-row cofactors are,

$A_{11}=\left[\begin{array}{l}\text{0 0}\\ \text{0 1}\end{array}\right]$

So, the determinant is obtained as,

$\begin{array}{l}{A}_{11}=0(1)-0(0)\\ =0\end{array}$

The second-row cofactors are,

$A_{12}=\left[\begin{array}{l}\text{1 0}\\ \text{0 1}\end{array}\right]$

So, the determi