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Chapter 3: Problem 8

Evaluate. \(\left|\begin{array}{rr}{3} & {2} \\ {2} & {-3}\end{array}\right|\)

Short Answer

Expert verified
The determinant is -13.

Step by step solution

01

Identify the elements of the matrix

Identify the elements a, b, c, and d from the given matrix elements: a = 3,b = 2,c = 2,d = -3
02

Apply the determinant formula

Use the formula for the determinant of a 2x2 matrix:determinant = ad - bc
03

Substitute the elements

Substitute the identified elements into the formula determinant = (3)(-3) - (2)(2)
04

Calculate the result

Perform the multiplications and subtraction:determinant = -9 - 4 which simplifies to determinant = -13

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Key Concepts

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

2x2 matrix
A 2x2 matrix is a simple and commonly used format in matrix algebra. It consists of two rows and two columns, forming a square. In general, a 2x2 matrix can be represented as follows:

\[ \begin{pmatrix} a & b \ c & d \ \end{pmatrix} \]
Here, 'a', 'b', 'c', and 'd' are the elements of the matrix. The rows are horizontal sequences of elements, and the columns are vertical sequences. This type of matrix is often the starting point for learning about matrices due to its simplicity and ease of manipulation.
Determinant formula
The determinant of a 2x2 matrix is a special number that can be calculated using a straightforward formula. For a matrix

\[ \begin{pmatrix} a & b \ c & d \ \end{pmatrix} \]
the determinant is calculated using the following formula:

\text{Determinant} = ad - bc
This formula gives a scalar value that has numerous implications in matrix algebra, such as determining if a matrix is invertible. Let's illustrate this with an example: for the matrix
\begin{pmatrix} 3 & 2 \ 2 & -3 \ \end{pmatrix},
the elements are identified as a = 3, b = 2, c = 2, and d = -3. By plugging these into the determinant formula, we have:

(3 \times -3) - (2 \times 2) \ = -9 - 4 \ = -13.
Matrix algebra
Matrix algebra involves operations with matrices, such as addition, multiplication, and finding determinants. It provides a powerful framework for solving systems of linear equations and performing various transformations. Here's a brief overview of common operations:

  • Matrix Addition: Adding corresponding elements from two matrices.
  • Scalar Multiplication: Multiplying every element of a matrix by the same scalar value.
  • Matrix Multiplication: Combining two matrices by multiplying rows of the first matrix by columns of the second.
  • Determinants: A value calculated from a square matrix that can help in understanding matrix properties, such as invertibility.


Matrix algebra is fundamental in various fields including physics, computer science, and engineering, as it allows for efficient handling and transformation of data.

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Most popular questions from this chapter

Solve each system. If a system's equations are dependent or if there is no solution, state this. $$ \begin{aligned} u-v+6 w &=8 \\ 3 u-v+6 w &=14 \\ -u-2 v-3 w &=7 \end{aligned} $$

Exercise. Huan jogs and walks to school each day. She averages \(4 \mathrm{km} / \mathrm{h}\) walking and \(8 \mathrm{km} / \mathrm{h}\) jogging. From home to school is \(6 \mathrm{km}\) and Huan makes the trip in 1 hr. How far does she jog in a trip?

Solve each system. If a system's equations are dependent or if there is no solution, state this. $$ \begin{aligned} x+y+z &=57 \\ -2 x+y &=3 \\ x-& z=6 \end{aligned} $$

Solve each system. If a system's equations are dependent or if there is no solution, state this. $$ \begin{aligned} 3 x+4 y-3 z &=4 \\ 5 x-y+2 z &=3 \\ x+2 y-z &=-2 \end{aligned} $$

Solve each system. If a system's equations are dependent or if there is no solution, state this. $$ \begin{array}{r} {-2 x+8 y+2 z=4} \\ {x+6 y+3 z=4} \\ {3 x-2 y+z=0} \end{array} $$
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