Problem 20 - Precalculus: Mathematical for Calculus

Question

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$\left[\begin{array}{rrr} 1 & 2 & 5 \\ -2 & -3 & 2 \\ 3 & 5 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 0; the matrix does not have an inverse.

Step 1: Matrix Determinant Formula for 3x3

To find the determinant of a 3x3 matrix \( A \) with elements \( a_{ij} \), use the formula: \[ \text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \] where \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \).

Step 2: Substitute the Matrix Elements

Substitute the elements of the matrix \( A = \begin{bmatrix} 1 & 2 & 5 \ -2 & -3 & 2 \ 3 & 5 & 3 \end{bmatrix} \) into the formula: \[ a = 1, b = 2, c = 5, d = -2, e = -3, f = 2, g = 3, h = 5, i = 3 \].

Step 3: Calculate Individual Terms

Calculate the three main terms of the determinant formula: 1. \((ei − fh) = (-3)(3) - (2)(5) = -9 - 10 = -19\)2. \((di − fg) = (-2)(3) - (2)(3) = -6 - 6 = -12\)3. \((dh − eg) = (-2)(5) - (-3)(3) = -10 + 9 = -1\)

Step 4: Apply and Simplify the Formula

Substitute the calculations back into the determinant formula: \[ \text{det}(A) = 1(-19) - 2(-12) + 5(-1) \]Simplifying gives: \[ -19 + 24 - 5 = 0 \]

Step 5: Determine Matrix Invertibility

A matrix is invertible if its determinant is not zero. Since \( \text{det}(A) = 0 \), matrix \( A \) is not invertible.

Key Concepts

Matrix Inversion

Matrix inversion is a process where we find a specific matrix known as the "inverse" of a given matrix. Have you ever thought about reversing an operation? That's what finding an inverse is all about. The inverse matrix, when multiplied by the original matrix, results in the identity matrix.

The identity matrix acts like the number 1 in multiplication. It doesn’t change the other number when you multiply by it. In the context of matrices, it's a square array with 1's along its diagonal and 0's everywhere else.

A matrix must be square (same number of rows and columns) and must have a non-zero determinant to be invertible, meaning that not all matrices have inverses.

• If the determinant of a matrix is zero, the matrix is said to be singular, and it cannot be inverted.
• If the determinant is not zero, the matrix is non-singular, meaning an inverse exists.

3x3 Matrix

A 3x3 matrix is simply a matrix with three rows and three columns, much like a small, organized grid of numbers. It's quite common when dealing with situations that require more complex mathematical operations, like graphics transformations or solving systems of equations.

Each element in this matrix is located at a specific row and column intersection, often represented by the notation \(a_{ij}\) where \(i\) is the row number and \(j\) is the column number.

When finding properties like the determinant, it's essential to follow specific steps and employ predefined formulas suitable for this square layout of numbers. Although the actual calculation can seem intricate, breaking it down into steps helps simplify the process tremendously.

Inverse Matrix Criteria

The criteria for a matrix to have an inverse revolve primarily around its determinant. If you're looking to learn how to verify whether a matrix is invertible, start by calculating its determinant.

Here's what you need to remember:

• Determinant Non-Zero: If the determinant is anything other than zero, the matrix can have an inverse.
• Determinant Zero: If the determinant is zero, this matrix is described as singular, indicating it doesn't have an inverse.

Additionally, ensuring that you have a square matrix is crucial because non-square matrices (like 2x3 or 3x2) don’t qualify for inverse calculation regardless of their determinants. This fundamental rule simplifies determining invertibility significantly, by only needing to focus on square matrices.