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Chapter 12: Problem 9

Find the value of each determinant. \(\left|\begin{array}{rr}-3 & -1 \\ 4 & 2\end{array}\right|\)

Short Answer

Expert verified
The determinant is -2.

Step by step solution

01

Understand the formula for 2x2 determinant

The formula for finding the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by: \[ \text{Determinant} = ad - bc \]
02

Identify the elements of the matrix

From the given matrix \(\begin{bmatrix} -3 & -1 \ 4 & 2 \end{bmatrix}\), identify the elements: \ a = -3, \ b = -1, \ c = 4, \ d = 2. \
03

Substitute the elements into the formula

Substitute the identified elements into the determinant formula: \[ \text{Determinant} = (-3)(2) - (-1)(4) \]
04

Perform the calculations

Calculate the values step by step: \[ \text{-3} \times 2 = -6 \] \[ \text{-1} \times 4 = -4 \] \[ -6 - (-4) = -6 + 4 = -2 \]
05

State the determinant

The determinant of the given matrix is \[ -2 \]

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Key Concepts

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

Matrix Determinant
The concept of a matrix determinant is crucial in linear algebra. A determinant is a special number that can be calculated from the elements of a square matrix. It provides important properties regarding the matrix, such as whether the matrix is invertible. For a 2x2 matrix, the determinant gives a quick way to determine the linear dependency of its columns.
Linear Algebra
Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. Matrices are a key element in this field. They are used to represent linear transformations and can be manipulated to understand more about these transformations. The determinant is one of the many tools in linear algebra to explore properties like eigenvalues, areas, and volumes.
2x2 Matrix
A 2x2 matrix consists of four elements arranged in two rows and two columns. Each element can be referred to by its position \((row, column)\). The determinant of a 2x2 matrix \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \] is calculated using the formula: \[ \text{Determinant} = ad - bc \] The given example demonstrates this clearly by identifying the matrix elements \(-3, -1, 4, 2\) and substituting them into the formula, yielding a determinant of \(-2\).

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