Chapter B, Problem 1DE

Evaluate.

$\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right]$

To determine

To calculate: The value of determinant $\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|$.

Answer to Problem 1DE

Solution:

The value of the determinant $\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|$ is $\underset{\_}{-5}$.

Explanation of Solution

Given information:

The provided second order determinant is $\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|$.

Formula used:

The determinant formula as:

$\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|=\left(a_{11}\right)\cdot \left(a_{22}\right)-\left(a_{12}\right)\cdot \left(a_{21}\right)$

Calculation:

Consider the determinant,

$\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|$

Use the formula to determine the value of the determinant;

$\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|=\left(a_{11}\right)\cdot \left(a_{22}\right)-\left(a_{12}\right)\cdot \left(a_{21}\right)$

Therefore,

$\begin{array}{c}\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|=\left(3\right)\cdot \left(1\right)-\left(2\right)\cdot \left(4\right)\\ =3-8\\ =-5\end{array}$

Hence, the value of the determinant $\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|$ is $-5$.