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Chapter 6: Problem 73

What happens to the value of a second-order determinant if the two columns are interchanged?

Short Answer

Expert verified
When the two columns of a second-order determinant are interchanged, the value of the determinant is multiplied by -1.

Step by step solution

01

Determine the determinant of the first order

Let's consider a second-order determinant with elements \(a, b, c, d\). The expression for the determinant is \(By = ad - bc\)
02

Interchange the columns

When the columns of the matrix are switched, the positions of \(b\) and \(c\) are swapped, and the new matrix has elements \(b, a, d, c\)
03

Determine the determinant of the new order

Now calculate the new determinant \(By' = ac - bd\)
04

Compare the determinants

By comparing the determinants \(By\) and \(By'\), it is observed that we obtain the negative of the original determinant when we switch the two columns of a second-order determinant. This happens because the process of switching columns is equivalent to multiplying the determinant by -1

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Most popular questions from this chapter

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows If an operation is not defined, state the reason. $$A=\left[\begin{array}{rr}4 & 0 \\\\-3 & 5 \\\0 & 1\end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\\\-2 & -2\end{array}\right] \quad C=\left[\begin{array}{rr}1 & -1 \\\\-1 & 1\end{array}\right]$$ $$4 B-3 C$$

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$\left[\begin{array}{rrrr}7 & -3 & 0 & 2 \\\\-2 & 1 & 0 & -1 \\\4 & 0 & 1 & -2 \\\\-1 & 1 & 0 & -1\end{array}\right]$$

If \(I\) is the multiplicative identity matrix of onder \(2,\) find \((I-A)^{-1}\) for the given matrix \(A\) $$\left[\begin{array}{rr}7 & -5 \\\\-4 & 3\end{array}\right]$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Give an example of a \(2 \times 2\) matrix that is its own inverse.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I added matrices of the same order by adding corresponding elements
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