Problem 37 - Elementary and Intermediate Algebra

Question

Solve. $$ \left|\begin{array}{rr} {m+1} & {-2} \\ {m-2} & {1} \end{array}\right|=27 $$

Step-by-Step Solution

Answer

m = 10

Step 1: Recognize the Determinant Formula for a 2x2 Matrix

For a 2x2 matrix \[ \begin{array}{cc} a & b \ c & d \end{array} \] , the determinant is calculated using the formula: \( \text{det} = ad - bc \).

Step 2: Identify the Elements of the Matrix

The matrix provided is: \[\begin{array}{rr} m+1 & -2 \ m-2 & 1 \end{array} \]. Here, \(a = m+1\), \(b = -2\), \(c = m-2\), and \(d = 1\).

Step 3: Substitute the Elements into the Determinant Formula

Substitute the values into the determinant formula: \( \text{det} = (m+1) \times 1 - (-2) \times (m-2) \).

Step 4: Simplify the Expression

Simplify the expression: \( \text{det} = m + 1 + 2(m-2) \). Combine like terms to get: \( \text{det} = m + 1 + 2m - 4 = 3m - 3 \).

Step 5: Set the Determinant Equal to 27

Set up the equation based on the determinant provided: \( 3m - 3 = 27 \).

Step 6: Solve for m

Add 3 to both sides to get: \( 3m = 30 \). Then divide both sides by 3 to find: \( m = 10 \).

Key Concepts

determinant of a 2x2 matrix

To start with, let's understand what a determinant is in the context of a 2x2 matrix. The determinant is a special number that can be calculated from the elements of a square matrix. For a 2x2 matrix formed by elements a, b, c, and d, we denote it as:
\begin{array}{cc} a & b \ c & d \ \text{end{array}The formula for calculating the determinant (det) is:
\[{det} = ad - bc \]
This formula gives us a scalar value that characterizes the matrix.
In our exercise, we have a matrix:
\begin{array}{rr} m+1 & -2 \ m-2 & 1\text{end{array}
To get the determinant of this matrix, we need to substitute the elements into our formula:
\[{det} = (m+1) \times 1 - (-2) \times (m-2)\]
which simplifies progressively to
\[det = 3m - 3 \]
and further used to solve the provided equation.

solving linear equations

Understanding how to solve linear equations is fundamental in algebra. Linear equations are those which graph as straight lines and can generally be written in the form ax + b = c.
In our problem, after simplifying the determinant, we have the linear equation: \[ 3m - 3 = 27 \].
To solve for m, follow these steps:

• Add 3 to both sides of the equation: 3m - 3 + 3 = 27 + 3
  
• Simplify: 3m = 30
  
• Divide both sides by 3: m = 30/3 \[ \Rightarrow m = 10 \]

By isolating the variable m, we find its value. This process shows how solving linear equations can be direct and structured.

matrix algebra

Matrix algebra is a branch of mathematics that deals with matrices and their operations. It's widely used in various fields such as computer graphics, physics, and economics. Working with matrices involves operations like addition, subtraction, multiplication, and finding determinants.
Matrices are organized in rows and columns, and their elements can be numbers, variables, or more complex expressions.
In our problem, understanding the given 2x2 matrix's elements was crucial to using the determinant formula: \[ \begin{array}{rr} m+1 & -2 \ m-2 & 1 \text{end{array}\] Matrix algebra simplified our approach to solving for m by breaking the problem into manageable steps: calculating the determinant and then solving the equation. By mastering these operations, you can tackle more complex matrix problems with confidence.