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Chapter 6: Problem 4

Evaluate each determinant. $$\left|\begin{array}{rr}7 & 9 \\\\-2 & -5\end{array}\right|$$

Short Answer

Expert verified
The determinant of the given matrix is -17.

Step by step solution

01

Identifying Elements of the Matrix

The given matrix is \[\left|\begin{array}{rr}7 & 9 \\-2 & -5\end{array}\right|\] Here, \(a = 7\), \(b = 9\), \(c = -2\), and \(d = -5\).
02

Applying the Determinant Formula

The determinant of a 2x2 matrix is calculated by the formula \(ad - bc\). Substituting the values from the matrix, we get \(7*(-5) - 9*(-2)\).
03

Solving the Expression

After doing the multiplications, the expression becomes \(-35 - (-18)\), which simplifies further to \(-35 + 18 = -17\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Determinants
Matrix determinants are fundamental in linear algebra, playing a crucial role in solving linear systems, understanding geometric properties, and more. A determinant provides a single numerical value that represents certain properties of a matrix.

Determinants provide insights into matrices by:
  • Determining whether a matrix is invertible
  • Indicating the volume scaling factor for linear transformations
  • Facilitating calculation of eigenvalues
Knowing how to calculate determinants is essential for anyone dealing with matrices, whether in mathematics, engineering, or computer science.
2x2 Matrix
A 2x2 matrix is one of the simplest forms of a matrix, consisting of two rows and two columns. Matrices are rectangular arrays of numbers, and in a 2x2 matrix, you will see something like this: \[\begin{array}{cc}a & b \c & d\end{array}\]

The elements in a 2x2 matrix are arranged in a very ordered structure:
  • Top left is `a`
  • Top right is `b`
  • Bottom left is `c`
  • Bottom right is `d`
This setup makes it particularly straightforward to apply formulas, such as the determinant formula. Simple matrix structures are often easier to manipulate and understand.
Determinant Formula
The determinant formula for a 2x2 matrix is a direct and simple calculation. Given a matrix \[\begin{array}{cc}a & b \c & d\end{array}\]

The determinant, denoted as \(det(A)\) or \(\left| A \right|\), can be determined using:
  • Multiply `a` and `d` to get \(ad\)
  • Multiply `b` and `c` to get \(bc\)
  • Subtract the product \(bc\) from the product \(ad\) to get \(ad - bc\)
Using this formula, you can find the determinant of any 2x2 matrix quickly. For example, in our exercise, it simplifies to substitute the numbers and solve: \(7\times (-5) - 9\times (-2)\), yielding a determinant of \(-17\). This succinct calculation reflects both the simplicity and utility of the determinant formula in linear algebra.

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Most popular questions from this chapter

Describe how to subtract matrices.

Exercises \(85-87\) will help you prepare for the material covered in the next section. Use Gauss-Jordan elimination to solve the system: $$\left\\{\begin{array}{cc}-x-y-z=1 \\\4 x+5 y & =0 \\\y-3 z=0\end{array}\right.$$

will help you prepare for the material covered in the next section. Simplify the expression in each exercise. $$2(-5)-(-3)(4)$$

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. $$A=\left[\begin{array}{rr}4 & 0 \\\\-3 & 5 \\\0 & 1\end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\\\-2 & -2\end{array}\right] \quad C=\left[\begin{array}{rr}1 & -1 \\\\-1 & 1\end{array}\right]$$ $$B C+C B$$

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