Chapter 7.2, Problem 54E

Solution

To determine: The definition given for the determinant of a square matrix agrees with the definition given of the determinant of a $2\times 2$ .

The determinant is same for square matrix as per the given condition.

Given Information:

The definition given for the determinant of a square matrix agrees with the definition given of the determinant of a $2\times 2$ .

Explanation:

Consider the given information,

The general matrix of order 2 is defined as,

  $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

As the above matrix is square matrix this implies that the matrix have equal row and column, which is 2. So from the definition of the determinant, the matrix should be square matrix to find the determinant.

And,

  $A_{11}=a,A_{12}=b,A_{21}=c,A_{22}=d$

So, apply the definition of determinate and calculate by using the first row.

  $\begin{array}{c}\left|A\right|=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|\\ =a_{11}{A}_{22}+a_{12}{A}_{21}\\ =a{\left(-1\right)}^2\left(d\right)+b{\left(-1\right)}^{3}c\\ =ad-bc\end{array}$

Apply the definition of determinate and calculate by using the second row.

  $\begin{array}{c}\left|A\right|=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|\\ =a_{21}{A}_{12}+a_{22}{A}_{11}\\ =c{\left(-1\right)}^3\left(b\right)+d{\left(-1\right)}^{4}a\\ =-bc+ad\end{array}$

Apply the definition of determinate and calculate by using the first column.

  $\begin{array}{c}\left|A\right|=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|\\ =a_{11}{A}_{22}+a_{21}{A}_{12}\\ =a{\left(-1\right)}^{2}d+c{\left(-1\right)}^{3}b\\ =ad-bc\end{array}$

Apply the definition of determinate and calculate by using the second column.

  $\begin{array}{c}\left|A\right|=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|\\ =a_{12}{A}_{21}+a_{22}{A}_{11}\\ =b{\left(-1\right)}^{3}c+d{\left(-1\right)}^{4}a\\ =-bc+ad\end{array}$

Therefore, the determinant is same for square matrix as per the given condition.