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Chapter 9: Problem 55

Explain how to evaluate a second-order determinant.

Short Answer

Expert verified
To evaluate a second-order determinant, identify the elements of the matrix, then apply the formula ad - bc.

Step by step solution

01

Identify the elements

In a 2x2 matrix, four elements can be identified. These are denoted as simply 'a', 'b', 'c', and 'd'. The matrix has the form: \[ \begin{matrix} a & b \ c & d \end{matrix} \].
02

Apply the formula

Once the elements of the matrix have been identified, apply the formula for a second-order determinant, which is ad - bc.
03

Calculate the Result

Perform the multiplication operations first, then subtract the results in accordance with the formula. This will yield the value of the second-order determinant.

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Most popular questions from this chapter

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{c} {w+x+y+z=5} \\ {w+2 x-y-2 z=-1} \\ {w-3 x-3 y-z=-1} \\ {2 w-x+2 y-z=-2} \end{array}\right. $$

If \(A B=-B A,\) then \(A\) and \(B\) are said to be anticommutative. Are \(A=\left[\begin{array}{rr}{0} & {-1} \\ {1} & {0}\end{array}\right]\) and \(B=\left[\begin{array}{rr}{1} & {0} \\ {0} & {-1}\end{array}\right]\) anticommutative?

Let $$ \begin{aligned} A &=\left[\begin{array}{cc} {1} & {0} \\ {0} & {1} \end{array}\right], \quad B=\left[\begin{array}{rr} {1} & {0} \\ {0} & {-1} \end{array}\right], \quad C=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {1} \end{array}\right] \\ D &=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {-1} \end{array}\right] \end{aligned} $$ Find the product of the difference between A and B and the sum of C and D.

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

Let $$ \begin{aligned} A &=\left[\begin{array}{cc} {1} & {0} \\ {0} & {1} \end{array}\right], \quad B=\left[\begin{array}{rr} {1} & {0} \\ {0} & {-1} \end{array}\right], \quad C=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {1} \end{array}\right] \\ D &=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {-1} \end{array}\right] \end{aligned} $$ Use any three of the matrices to verify a distributive property.
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