Chapter 15.1, Problem 56E

To determine

To find the value of the given determinanats.

Answer to Problem 56E

The vaule of determinant is $12$

Explanation of Solution

Given:

The matrix

  $\left[\begin{array}{cc}x& x\ln x\\ 1& 1+\ln x\end{array}\right]$

Concept used:

Definition of matrix multiplication :-if A $=\left[a_{ij}\right]$ is an $m\times n$ matrix and B $=\left[b_{ij}\right]$ is an $n\times p$ matrix , then the product $AB$ is an $m\times p$ matrix

  $\begin{array}{l}AB=\left[a_{ij}\times b_{ij}\right]\\ AB=\left[c_{ij}\right]\end{array}$

Where $c_{ij}$ = $=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}+\mathrm{..........}+a_{in}{b}_{nj}$

Calculation:

The product $AB$ is an $m\times p$ matrix

  $\begin{array}{l}AB=\left[a_{ij}\times b_{ij}\right]\\ AB=\left[c_{ij}\right]\end{array}$

Where $c_{ij}$ = $=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}+\mathrm{..........}+a_{in}{b}_{nj}$

The determinant

  $\begin{array}{l}\left|\begin{array}{cc}-1& -2\\ 3& -6\end{array}\right|\\ 6+6\\ 12\end{array}$

The vaule of determinant is $12$