Problem 30 - A Graphical Approach to Precalculus with Limits

Question

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{rrr} 5 & -3 & 2 \\ -5 & 3 & -2 \\ 1 & 0 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix \(A\) does not have an inverse because its determinant is 0.

Step 1: Calculate determinant of A

To find if the inverse of matrix \(A\) exists, we need to calculate its determinant. The determinant of a 3x3 matrix \(A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \) is given by:\[ \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]Substitute the given values:\[ \text{det}(A) = 5(3 \times 1 - (-2) \times 0) - (-3)(-5 \times 1 - 1 \times (-2)) + 2((-5) \times 0 - 3 \times 1) \]\[ = 5(3) - (-3)(-5 + 2) + 2(0 - 3) \]\[ = 15 - 9 + (-6) \]\[ = 0 \]The determinant is 0.

Step 2: Determine inverse existence

A matrix can only have an inverse if its determinant is non-zero. Since the determinant of matrix \(A\) is 0, matrix \(A\) does not have an inverse.

Key Concepts

Determinant Calculation

The determinant is a special number that can be calculated from a square matrix. It provides essential information about the matrix, particularly regarding its invertibility. To calculate the determinant of a 3x3 matrix, you use the formula:
\[ \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\]where the matrix \[A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \]In simpler terms:

• Each element of the first row is multiplied by the determinant of a 2x2 matrix formed by eliminating the row and column of that element.
• Then, these products are summed, while alternately adding and subtracting them.

For example, using the matrix from the exercise, we calculated:
\[ \text{det}(A) = 5(3 \times 1 - (-2) \times 0) - (-3)(-5 \times 1 - 1 \times (-2)) + 2((-5) \times 0 - 3 \times 1) \]which simplifies to:
\[ 15 - 9 + (-6) = 0 \]A determinant of zero indicates that the matrix is singular, meaning it does not have an inverse.

3x3 Matrix

A 3x3 matrix is a mathematical representation that has three rows and three columns. It's commonly used in various applications like engineering, physics, and computer science. The format of a 3x3 matrix is typically expressed as:
\[ A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \]Each element of the matrix is denoted by a letter like \(a, b, c,\) etc. These elements can be numbers, variables, or expressions.

• The first row is \([a, b, c]\).
• The second row is \([d, e, f]\).
• The third row is \([g, h, i]\).

Understanding a 3x3 matrix involves identifying its elements and how they interact in operations like addition, subtraction, and multiplication.
3x3 matrices are particularly vital because they allow for transformations in 3-dimensional space. They can represent systems of equations or transformations like rotations and scaling in graphics.

Inverse of a Matrix

The inverse of a matrix \(A^{-1}\) is a matrix that, when multiplied with the original matrix \(A\), results in the identity matrix. For a matrix to have an inverse, its determinant must not be zero. This is because the inverse is calculated using the determinant.
The process of finding the inverse involves several steps for a 3x3 matrix, including:

• Calculating the determinant to ensure it's non-zero.
• Using the elements of the matrix to find the adjugate (or adjoint) matrix.
• Dividing the adjugate by the determinant.

If a matrix is singular, like the one in the exercise with a determinant of zero, it means that the matrix cannot be inverted. This happens because any number divided by zero is undefined in mathematics.
Understanding when a matrix has an inverse is crucial for solving equations that use matrices, such as in linear algebra problems.