Problem 24 - Precalculus: Mathematics for Calculus

Question

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$\left[\begin{array}{rrr} -2 & -\frac{3}{2} & \frac{1}{2} \\ 2 & 4 & 0 \\ \frac{1}{2} & 2 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is -4; the matrix is invertible.

Step 1: Identify the Matrix Components

We are given a 3x3 matrix: \[\begin{bmatrix} -2 & -\frac{3}{2} & \frac{1}{2} \ 2 & 4 & 0 \ \frac{1}{2} & 2 & 1 \end{bmatrix} \] Our task is to find the determinant of this matrix, which will help us determine if the matrix is invertible.

Step 2: Recall the Determinant Formula for 3x3 Matrices

The formula for the determinant of a 3x3 matrix \( A \) is given by: \[\det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]where the matrix is structured as: \[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]

Step 3: Assign Elements from Matrix to Variables

In the given matrix:\[\begin{bmatrix} -2 & -\frac{3}{2} & \frac{1}{2} \ 2 & 4 & 0 \ \frac{1}{2} & 2 & 1 \end{bmatrix} \]we assign: \( a = -2 \), \( b = -\frac{3}{2} \), \( c = \frac{1}{2} \), \( d = 2 \), \( e = 4 \), \( f = 0 \), \( g = \frac{1}{2} \), \( h = 2 \), \( i = 1 \).

Step 4: Calculate the Determinant

Substitute the assigned values into the determinant formula:\[\det(A) = -2(4 \times 1 - 0 \times 2) - \left(-\frac{3}{2}\right)(2 \times 1 - 0 \times \frac{1}{2}) + \frac{1}{2}(2 \times 2 - 4 \times \frac{1}{2}) \]This simplifies to:\[\det(A) = -2(4) + \frac{3}{2}(2) + \frac{1}{2}(4 - 2) \]which further simplifies to:\[\det(A) = -8 + 3 + 1 = -4 \]

Step 5: Determine Invertibility

The determinant of the matrix is \(-4\). Since the determinant is not zero, the matrix is invertible.

Key Concepts

Matrix Inverse

A matrix inverse is a vital concept in linear algebra. It involves finding another matrix, known as the inverse matrix, which when multiplied with the original matrix, yields the identity matrix. To find out if a matrix is invertible, we first need to determine its determinant.
For a matrix to have an inverse, its determinant must be non-zero. Thus, calculating the determinant helps us to check the invertibility without actually finding the inverse.
Here's how it works:

• If the determinant of the matrix is zero, the matrix does not have an inverse and is termed singular.
• Conversely, if the determinant is non-zero, the matrix is invertible or non-singular.

Finding an inverse involves more processes but always starts with checking the determinant, as it gives us the basic information needed.

3x3 Matrix

A 3x3 matrix, sometimes referred to as a square matrix, consists of three rows and three columns. This type of matrix is often encountered in various mathematical computations in linear algebra.
Because of its structure, specific rules and formulas apply to 3x3 matrices. One particularly important calculation is finding the determinant, which can be thought of as a “special number” associated with a matrix.
To compute the determinant of a 3x3 matrix, we use the formula:
\[ ext{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]Each variable in the formula corresponds to a specific position in the matrix, allowing for a precise calculation based on the arrangement of numbers.

• A correct determinant calculation confirms if the matrix is invertible.
• Familiarity with the structure of a 3x3 matrix facilitates such calculations.

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, vector spaces, and linear transformations. It is fundamental in understanding how we perform operations on matrices, analyze their properties, and solve related problems.
Matrices are central objects of study in linear algebra. They help in representing and solving systems of linear equations.
Key concepts in linear algebra include:

• Systems of Equations: Linear algebra provides methods such as Gaussian elimination to find solutions.
• Matrix Operations: Multiplication, addition, and finding inverses are central operations.
• Determinant: Useful in determining properties like invertibility of a matrix.

By mastering linear algebra, one gains the ability to handle complex problems in various fields, from engineering to data science.