Problem 25 - Precalculus: Mathematics for Calculus

Question

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$\left[\begin{array}{lll} 1 & 3 & 7 \\ 2 & 0 & 8 \\ 0 & 2 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 0, so the matrix has no inverse.

Step 1: Identify the matrix elements

First, write down the given 3x3 matrix and note its elements: \[\mathbf{A} = \begin{bmatrix} 1 & 3 & 7 \ 2 & 0 & 8 \ 0 & 2 & 2 \end{bmatrix}\]. The elements are arranged as follows: row 1 = [1, 3, 7], row 2 = [2, 0, 8], and row 3 = [0, 2, 2].

Step 2: Use the formula for a 3x3 determinant

For a matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant is calculated as follows: \[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\].

Step 3: Substitute the matrix elements into the determinant formula

Using the elements of the given matrix: \( a = 1 \), \( b = 3 \), \( c = 7 \), \( d = 2 \), \( e = 0 \), \( f = 8 \), \( g = 0 \), \( h = 2 \), \( i = 2 \), substitute into the formula:\[\text{det}(A) = 1(0 \cdot 2 - 8 \cdot 2) - 3(2 \cdot 2 - 8 \cdot 0) + 7(2 \cdot 2 - 0 \cdot 0)\].

Step 4: Calculate each term in the formula

Compute each term:1. The first term is \( 1(0 - 16) = 1(-16) = -16 \).2. The second term is \( 3(4 - 0) = 3(4) = 12 \).3. The third term is \( 7(4 - 0) = 7(4) = 28 \).

Step 5: Sum the terms to find the determinant

Add the terms from the previous step:\[\text{det}(A) = -16 - 12 + 28 = 0\]. The determinant of the matrix is 0.

Step 6: Determine if the matrix has an inverse

A matrix is invertible if its determinant is non-zero. Since the determinant of this matrix is 0, it does not have an inverse.

Key Concepts

Invertibility

In the world of matrices, invertibility is a vital concept. A matrix is said to be invertible if there exists another matrix that, when multiplied with the original matrix, results in the identity matrix. The identity matrix acts like the number 1 in basic arithmetic, meaning it doesn't change the other matrix when multiplied.
To check for invertibility, we look at the determinant of the matrix. The rule is simple:

• If the determinant is non-zero, the matrix is invertible.
• If the determinant is zero, the matrix is not invertible.

For the given matrix, after calculating the determinant, we found it to be zero. This means that the matrix doesn't have an inverse. In other words, there is no matrix that can reverse its effects when multiplied together. Understanding and verifying invertibility is crucial because it influences whether certain matrix operations can be executed.

3x3 Matrix

Studying a 3x3 matrix is essential for understanding more complex linear algebra concepts. A 3x3 matrix is essentially a square grid with three rows and three columns. This layout holds nine elements in total, arranged in three rows and three columns.
In our example, the matrix is:\[\begin{bmatrix} 1 & 3 & 7 \ 2 & 0 & 8 \ 0 & 2 & 2 \end{bmatrix}\]The rows go horizontally, and the columns vertically. In practical terms, these matrices can be used to solve systems of equations with three variables. They also occur frequently in graphics, physics, and many other fields.
Understanding 3x3 matrices helps develop the fundamental skills required to manipulate larger systems and solve complex problems, setting a strong foundation for more advanced mathematics.

Determinant Calculation

Calculating the determinant is crucial in linear algebra as it helps us understand the properties of a matrix. For a 3x3 matrix like the one in our exercise, we use a specific formula:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]Where each letter corresponds to an element in the matrix:

• \(a, b, c\) from the first row
• \(d, e, f\) from the second row
• \(g, h, i\) from the third row

Our matrix:\[\begin{bmatrix} 1 & 3 & 7 \ 2 & 0 & 8 \ 0 & 2 & 2 \end{bmatrix}\]Substituting the given values yields:\[\text{det}(A) = 1(0 \cdot 2 - 8 \cdot 2) - 3(2 \cdot 2 - 8 \cdot 0) + 7(2 \cdot 2 - 0 \cdot 0)\]Step by step:

• First term: \(1(0 - 16) = -16\)
• Second term: \(3(4 - 0) = 12\)
• Third term: \(7(4 - 0) = 28\)

Adding them up gives us a determinant of zero. Accurate calculation of the determinant gives information not only about invertibility but also about linear transformations associated with the matrix.