Problem 22 - 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 has an inverse.

Step 1: Determine the Matrix Size

The given matrix is \( 3 \times 3 \), which means it has 3 rows and 3 columns. For a square matrix, the determinant can be calculated.

Step 2: Use the Determinant Formula for a 3x3 Matrix

For a matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant is \( a(ei - fh) - b(di - fg) + c(dh - eg) \).

Step 3: Plug Values into the Determinant Formula

Identify the elements of the matrix: \( a = -2 \), \( b = -\frac{3}{2} \), \( c = \frac{1}{2} \), \( d = 2 \), \( e = 4 \), \( f = 0 \), \( g = \frac{1}{2} \), \( h = 2 \), \( i = 1 \). Substitute these into the formula.

Step 4: Calculate Each Component

1. Find \( ei - fh = 4 \cdot 1 - 0 \cdot 2 = 4 \).2. Find \( di - fg = 2 \cdot 1 - 0 \cdot \frac{1}{2} = 2 \).3. Find \( dh - eg = 2 \cdot 2 - 4 \cdot \frac{1}{2} = 4 - 2 = 2 \).

Step 5: Calculate the Determinant

Substitute the calculated values from Step 4 into the determinant formula: \(-2(4) - \left(-\frac{3}{2}\right)(2) + \frac{1}{2}(2)\).Thus, the determinant is: \(-8 + 3 + 1 = -4\).

Step 6: Determine the Invertibility

A matrix is invertible if its determinant is non-zero. Since the determinant is \(-4\), which is not zero, the matrix has an inverse.

Key Concepts

Matrix Inversion

Matrix inversion is a fundamental concept in both mathematics and computer science. It's important because it allows you to "undo" the effect of a matrix on another matrix or vector.
Think of it as finding a reverse operation. Just like multiplying a number by its reciprocal gives you 1, multiplying a matrix by its inverse results in the identity matrix.

• The identity matrix is a special matrix where all the diagonal elements are 1, and all other elements are 0.
• Not all matrices can be inverted. A matrix has an inverse only if its determinant is non-zero.
• Inverting matrices is crucial for solving systems of linear equations. It helps us find solutions to equations of the form \( Ax = b \), by transforming it into \( x = A^{-1}b \).

So, you can only invert matrices that have a non-zero determinant, and by determining the determinant first, we ascertain whether the procedure can go ahead or not.

3x3 Matrix

A 3x3 matrix is a square matrix that has three rows and three columns. It's a common size for matrices used in various applications in science and engineering.
With nine elements, you can imagine them laid out neatly as a table with elements specified as \( a, b, c, d, e, f, g, h, i \).

• Calculating the determinant of a 3x3 matrix involves a specific formula. Each element plays a unique role in the computation.
• The determinant is crucial because it tells us about the matrix's properties, including whether it can be inverted.
• Finding the determinant involves summing the products of its elements, considering the positions and signs.

In practice, playing around with smaller matrices like 2x2 helps build the intuition, but 3x3 matrices are the next step where the concept solidifies.

Linear Algebra Concepts

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces.
Key concepts such as determinants and inverses of matrices emphasized in the exercise are foundational to mastering linear algebra.

• Matrices are a way to represent and solve linear equations. Everything in linear algebra boils down to handling these matrices effectively.
• Look for patterns and properties of matrices, like the determinant, which is a calculated number that tells us important information.
• Determinants specifically help tell if a matrix transformation is reversible, which is foundational for solving systems of linear equations.

By grasping these foundational elements, you put yourself in a strong position to understand more complex problems and solutions in linear algebra.