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Chapter 12: Problem 17

Evaluate each determinant. \(\left|\begin{array}{rr}-9 & 7 \\ 4 & -2\end{array}\right|\)

Short Answer

Expert verified
The determinant of the matrix is -10.

Step by step solution

01

Understand the Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated using the formula: \( ad - bc \). In this exercise, our matrix is \( \begin{pmatrix} -9 & 7 \ 4 & -2 \end{pmatrix} \).
02

Identify the Elements of the Matrix

Identify the elements \( a, b, c, \) and \( d \) from the matrix: \( a = -9 \), \( b = 7 \), \( c = 4 \), and \( d = -2 \).
03

Substitute the Elements into the Formula

Substitute the identified values into the formula for the determinant: \( ad - bc = (-9)(-2) - (7)(4) \).
04

Calculate the Determinant

Perform the calculations: First, multiply \( a \) and \( d \): \((-9) \times (-2) = 18\). Then multiply \( b \) and \( c \): \(7 \times 4 = 28\). The determinant is \( 18 - 28 = -10 \).

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Key Concepts

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

Matrix Operations
Matrix operations are fundamental tools in both mathematics and applied sciences. A matrix is essentially an array of numbers arranged in rows and columns. Matrix operations include addition, subtraction, and multiplication. One common operation is finding the *determinant* of a 2x2 matrix, which is a special value calculated from its elements.

For example, consider a 2x2 matrix:
  • The determinant is a singular number obtained from this arrangement.
  • This number helps to determine different properties like invertibility and eigenvalues.
The formula for the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is \( ad - bc \). This particular operation helps simplify more complex calculations in linear algebra.
Linear Algebra
Linear algebra is a branch of mathematics that focuses on vectors, matrices, and linear transformations. A key aspect of linear algebra is solving systems of linear equations, which often involves using matrices. One of the essential properties of a matrix is its determinant.

The determinant provides valuable insights into a matrix's behavior. For instance:
  • A non-zero determinant indicates that the matrix is invertible.
  • If a matrix's determinant equals zero, it means the matrix is singular and cannot be inverted.
Understanding these properties helps in examining the solutions of linear systems and studying vector spaces. It also lays the groundwork for more advanced topics like eigenvectors and eigenvalues.
Elementary Algebra
Elementary algebra introduces symbols and letters to represent numbers and quantities in formulas and equations. Matrices, a higher concept within this realm, enhance our ability to solve systems of equations efficiently.

In elementary algebra, working with a 2x2 matrix involves simple arithmetic operations such as:
  • Multiplying and summing the elements to compute the determinant.
  • Using basic formulas, such as \( ad - bc \), to find key values quickly.
These basic operations help build the foundation upon which more complex algebraic problems can be understood and solved. By understanding the determinant through elementary algebra, students develop the skills necessary to tackle larger, more complex matrices in higher mathematics.

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Most popular questions from this chapter

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 3 x-2 y=\frac{9}{2} \\ \frac{x}{2}-\frac{3}{4}=2 y \end{array}\right. $$

Which method would you use to solve the system? Explain. $$ \left\\{\begin{array}{l} y-1=3 x \\ 3 x+2 y=12 \end{array}\right. $$

Solve each system. To do so, substitute a for \(\frac{1}{x}\) and \(b\) for \(\frac{1}{y}\) and solve for a and \(b\). Then find \(x\) and \(y\) using the fact that \(a=\frac{1}{x}\) and \(b=\frac{1}{y}\) $$ \left\\{\begin{array}{l} \frac{1}{x}+\frac{1}{y}=\frac{9}{20} \\ \frac{1}{x}-\frac{1}{y}=\frac{1}{20} \end{array}\right. $$

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} a+\frac{b}{3}=\frac{5}{3} \\ \frac{a+b}{3}=3-a \end{array}\right. $$

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) $$ \left\\{\begin{array}{l} 4 x-3 y=5 \\ y=-2 x \end{array}\right. $$
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